Math Puzzle Wiki - User contributions [en]

Meta:Main Page

2024-08-18T17:43:17Z

Oscarlevin:

<div id="mf-home">
='''Welcome to the Math Puzzle Wiki'''=
[[File:SiteLogo.png|left]]
This is a collection of the best puzzles and brain teasers which contain some mathematical content. Currently there {{NUMBEROFARTICLES}} puzzles on the site. Go ahead and try your hand at a [[Special:Random|random puzzle]].

Many of the puzzles here are classics, although some are original. When possible, sources of puzzles are referenced. If you know the original source of a puzzle, please share.

Some puzzles already have hints, answers, and solutions. These should be hidden by default to avoid spoilers. For those that don't yet have solutions, why not add one to the site?

Want to [[Help:Ways to contribute|contribute]]? [[Special:UserLogin|Create and account or sign in]] and post your favorite puzzle.

<div style="clear:both; float: left; width: 49%">
==Types of puzzles==
<categorytree hideroot=on mode=all>Puzzle types</categorytree>

</div><div style="float: left; width: 49%; margin-left: 2%">
==Mathematical topics==
<categorytree hideroot=on mode=pages>Math topics</categorytree>
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<br style="clear: both; height: 0; line-height: 0; font-size: px; /*for IE*/"/>
See [[Special:Categories|All categories]] for more.

Check out the list of [[Special:AllPages | all puzzles]] for many more.

==References==

If you are looking for even more puzzles, check out our list of [[Help:References|Links]]

==Support Math Puzzle Wiki==

If you plan on buying something from Amazon, please link from [http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2F&tag=matpuzwik-20&linkCode=ur2&camp=1789&creative=390957 here]. Doing so will help with the costs of maintaining this site.

</div>
__NOTOC__

Flying trains

2023-10-05T14:53:03Z

Oscarlevin: /* Puzzle */

[[File:Train template.svg|right|150px]]

Here is the classic not-really-a-calculus puzzle.

==Puzzle==

Towns A and B are connected by a single railroad track, exactly 210 miles long. One fateful day, at exactly 1:00pm, a red train leaves town A traveling to town B at 40 miles per hour. At the same time, a bright blue train leaves town B traveling to town A at 30 miles per hour. As the red train starts to move, a brave fly takes off of the windshield and flies at 55 miles per hour towards town B. As soon as the fly reaches the blue train, he immediately changes direction and flies back towards town A, again, traveling at 55 miles per hour. When he gets to the red train, he changes direction again. The fly continues to fly back and forth between the two, ever nearing trains until he is smashed to bits when the trains sadly collide.

How far did the fly fly between 1:00pm and his all-to-early death?

==Help==

{{Hint | You certainly could try to set up some sort of infinite sum, but there is an easier way. First answer this: how ''long'' did the fly travel?}}
{{Answer | 165 miles.}}
{{Solution | Given the velocities of the trains, the distance between them is decreasing at a rate of 70 miles per hour. Thus the trains will collide in exactly 3 hours. The fly will be traveling at 55 miles per hour for this entire 3 hour period, which comes to 165 miles traveled.}}

==See also==

[[Girl, boy and dog]]

[[Category: Velocity puzzles]]
[[Category: Calculus]]

Ball drop

2019-09-03T21:33:48Z

Oscarlevin: /* Puzzle */

Here is a nice optimization puzzle from the [http://www.mathsisfun.com: Maths is Fun] website.

==Puzzle==

Pool and Billiard mega store Balls-R-Us wants to shoot a commercial for their new Nearly-Indestructible-Billiard-Balls-Are-Amazing brand billiard balls. The commercial will feature a crazed pool shark smashing billiard balls by dropping them from the top of a tall building, only to find that when he drops the NIBBAA balls, they don't break! To make the commercial as convincing as possible, the company wants to use as tall a building as possible, so they need to know the highest floor their billiard balls can be dropped from without breaking.

While you have a perfectly good 100 story building to test out the procedure, the producers of the commercial have only given you two billiard balls and want an answer as soon as possible. You realize you could test out each floor in order (first floor, then second floor, and so on), since you can reuse a ball that does not break. But that might take FOREVER! Is there a faster way? What is the least number of times you would have to drop the billiard balls to guarantee finding the highest safe floor?

==Variations==

Of course we could ask the same question using a different number of floors, or a different number of balls. In general, what is the least number of test drops needed to guarantee finding the highest safe floor when you have <m>n</m> balls and <m>m</m> total floors.

==Links==

[http://www.mathsisfun.com/puzzles/dropping-balls.html Dropping Balls] As described on Maths is Fun.

[[Category: Optimization puzzles]]
[[Category: Algorithms]]

Meta:Links

2019-07-18T04:00:37Z

Oscarlevin:

Looking for more puzzles? Here are some problem solving related resources.

==Collections and Blogs==

*[https://www.theguardian.com/science/series/alex-bellos-monday-puzzle Monday Puzzle Blog] by Alex Bellos.
*[https://www.popularmechanics.com/riddles-logic-puzzles/ Riddle of the Week] from Popular Mechanics.
*[https://blogs.wsj.com/puzzle/category/varsity-math-2/ Varsity Math] Puzzle block of the Wall Street Journal.
*[https://projecteuler.net/ Project Euler]. A series of challenging math and computer programming problems.
*[http://www.mathpuzzle.com/ MathPuzzle.com]
*[https://blog.tanyakhovanova.com/category/puzzles/ Tanya Khovanova's Math Blog Puzzles].
*{{Car Talk}}
*{{Mathisfun}}
*{{Mathcentral}}
*{{pzzls}}
*[https://www.unco.edu/nhs/mathematical-sciences/challenge/ The Math Challenge Problem] from the University of Norther Colorado.

==Videos==

*[https://www.youtube.com/playlist?list=PLJicmE8fK0EhMjOWNNhlY4Lxg8tupXKhC TED-Ed Riddles on YouTube]. Nice animations for classic puzzles.
*[https://www.youtube.com/user/MindYourDecisions/featured Mind Your Decisions]. See also the corresponding [https://mindyourdecisions.com/blog/ blog].
*[https://www.youtube.com/user/numberphile/featured Numberphile]. Lots of interesting math, including a number of good puzzles.

==Podcasts==

*{{Math Factor}} A math puzzle podcast from a while back.
*[https://www.npr.org/series/4473090/sunday-puzzle The Sunday Puzzle on NPR]. Mostly word puzzles, but occasionally an interesting math nugget.

==Books==

*{{Problem Solving}}
*{{Averbach}}
*{{Martin Gardner books}}
*{{Winkler book}}.
*{{Smullyan riddle}} Very nice collection of puzzles, all tied together in a story. Plus, some logic content as well.
*{{Sideways Arithmetic}} A collection of cryptarithmetic puzzles.

==Games==

*{{Mindtrap}}
*{{Professor Layton}}
*{{Professor Layton 2}}

==Problem Solving and Teaching Resources==

*[https://www.mathteacherscircle.org/resources/math-sessions/ Math Teacher Circles Sessions].

__NOTOC__

Meta:Links

2019-07-18T02:52:26Z

Oscarlevin:

Looking for more puzzles? Here are some problem solving related resources.

==Collections and Blogs==

*[https://www.theguardian.com/science/series/alex-bellos-monday-puzzle Monday Puzzle Blog] by Alex Bellos.
*[http://www.mathpuzzle.com/ MathPuzzle.com]
*{{Car Talk}}
*{{Mathisfun}}
*{{Mathcentral}}
*{{pzzls}}
*[https://www.unco.edu/nhs/mathematical-sciences/challenge/ The Math Challenge Problem] from the University of Norther Colorado.

==Videos==

*[https://www.youtube.com/playlist?list=PLJicmE8fK0EhMjOWNNhlY4Lxg8tupXKhC TED-Ed Riddles on YouTube]. Nice animations for classic puzzles.
*[https://www.youtube.com/user/MindYourDecisions/featured Mind Your Decisions]. See also the corresponding [https://mindyourdecisions.com/blog/ blog].
*[https://www.youtube.com/user/numberphile/featured Numberphile]. Lots of interesting math, including a number of good puzzles.

==Podcasts==

*{{Math Factor}} A math puzzle podcast from a while back.
*[https://www.npr.org/series/4473090/sunday-puzzle The Sunday Puzzle on NPR]. Mostly word puzzles, but occasionally an interesting math nugget.

==Books==

*{{Problem Solving}}
*{{Averbach}}
*{{Martin Gardner books}}
*{{Winkler book}}.
*{{Smullyan riddle}} Very nice collection of puzzles, all tied together in a story. Plus, some logic content as well.
*{{Sideways Arithmetic}} A collection of cryptarithmetic puzzles.

==Games==

*{{Mindtrap}}
*{{Professor Layton}}
*{{Professor Layton 2}}

==Problem Solving and Teaching Resources==

*[https://www.mathteacherscircle.org/resources/math-sessions/ Math Teacher Circles Sessions].

__NOTOC__

Template:Bradley PoW

2019-07-18T02:22:41Z

Oscarlevin:

Bradley University's Problem of the Week (no longer available).

Meta:Links

2019-07-18T02:21:23Z

Oscarlevin:

Looking for more puzzles? Here are some problem solving related resources.

==Websites==

*{{Math Factor}} A math puzzle podcast.
*{{Car Talk}}
*{{Mathisfun}}
*{{Mathcentral}}
*{{pzzls}}
*{{Richard Wiseman}}
*{{Bradley PoW}} No longer active, but there are 10 years worth of archives. From professor Alberto L. Delgado.

==Books==

*{{Problem Solving}}
*{{Averbach}}
*{{Martin Gardner books}}
*{{Winkler book}}.
*{{Smullyan riddle}} Very nice collection of puzzles, all tied together in a story. Plus, some logic content as well.
*{{Sideways Arithmetic}} A collection of cryptarithmetic puzzles.

==Games==

*{{Mindtrap}}
*{{Professor Layton}}
*{{Professor Layton 2}}

Ants on a cube

2017-10-26T15:15:32Z

Oscarlevin: Created page with "This is a generalization of the first puzzle in Presh Talwalkar's ''Math Puzzles Volume 1''. == Puzzle == 8 ants sit on the vertices of a cube (floating in space). Each ant..."

This is a generalization of the first puzzle in Presh Talwalkar's ''Math Puzzles Volume 1''.

== Puzzle ==

8 ants sit on the vertices of a cube (floating in space). Each ant sets off along one of the edges incident to its vertex, at random. What is the probability that no ants will collide?

Cents per cents

2017-10-01T19:15:04Z

Oscarlevin:

This is my version of a puzzle I originally saw on [https://www.theguardian.com/science/2017/jun/19/can-you-solve-it-pythagorass-best-puzzles Alex Bellos's Puzzle Blog].

==Puzzle==

100 of the dimmest bank robbers recently broke in to the Denver Mint and made off with a truck-load of fresh new pennies. To divvy up their loot, they decide that the youngest thief will get 1% of the score, then the 2nd youngest will get 2% of what is left, the third youngest getting 3% of what is left after that, and so on, until the oldest (100th youngest) gets 100% of what is left after everyone has taken their share.

Which thief receives the largest share? Further, what is the least amount of money he will leave with, assuming no rounding occurs while splitting the take?

Cents per cents

2017-08-06T14:32:28Z

Oscarlevin: Created page with "This is my version of a puzzle I originally saw on [https://www.theguardian.com/science/2017/jun/19/can-you-solve-it-pythagorass-best-puzzles Alex Bellos's Puzzle Block]. ==P..."

This is my version of a puzzle I originally saw on [https://www.theguardian.com/science/2017/jun/19/can-you-solve-it-pythagorass-best-puzzles Alex Bellos's Puzzle Block].

==Puzzle==

100 of the dimmest bank robbers recently broke in to the Denver Mint and made off with a truck-load of fresh new pennies. To divvy up their loot, they decide that the youngest thief will get 1% of the score, then the 2nd youngest will get 2% of what is left, the third youngest getting 3% of what is left after that, and so on, until the oldest (100th youngest) gets 100% of what is left after everyone has taken their share.

Which thief receives the largest share? Further, what is the least amount of money he will leave with, assuming no rounding occurs while splitting the take?

Next letter

2017-07-25T13:16:29Z

Oscarlevin: added links.

A nice short puzzle, perfect for posing at the start of class.

==Puzzle==

What are the next letters in the sequence: O T T F F S S E?

==Help==

{{Hint|The next letter is N}}

{{Answer|N T E T T ...}}

{{Solution|The letters in the sequence are the initials of the natural numbers: One, Two, Three, etc. So the sequence continues: N T E T T F F S S E N T.}}

==See Also==

[[Next sequence]]

[[Sequence next in sequence]]

[[Category: Lateral thinking]]
[[Category: Short puzzles]]

Next sequence

2017-07-25T13:14:21Z

Oscarlevin:

Here is a classic sequence guessing puzzle.

==Puzzle==

Consider the sequence 1, 11, 21, 1211, 111221, 312211... What comes next?

==Help==

{{Answer| 13112221.}}

==See Also==

[[Sequence next in sequence]]

[[Category: Lateral thinking]]
[[Category: Sequences]]
[[Category: Short puzzles]]

Sequence next in sequence

2017-07-25T13:10:46Z

Oscarlevin: Created page with "This sequence guessing puzzle hides a beautiful, well known sequence. ==Puzzle== What comes next: 1, 2, 2, 1, 1, 2, 1, ....? ==Help== {{Hint| Count the number of repeating..."

This sequence guessing puzzle hides a beautiful, well known sequence.

==Puzzle==

What comes next: 1, 2, 2, 1, 1, 2, 1, ....?

==Help==

{{Hint| Count the number of repeating digits.}}

{{Answer| 2}}

{{Solution| This is the famous Kolakoski sequence (see the wikipedia page listed below). The sequence self-encodes the run length of repeated digits. These run lengths are 1, 2, 2, 1, 1, 2, 1,...., which is the sequence again. }}

==Bonus==

Find a sequence using the numbers 1, 2, and 3 that has the same property as the one above.

==References==

[https://en.wikipedia.org/wiki/Kolakoski_sequence Wikipedia page] (contains spoilers).

[[Category:Short puzzles]]
[[Category:Sequences]]

__NOTOC__

Wagon collision

2017-07-25T02:19:31Z

Oscarlevin: /* Help */

This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

==Puzzle==

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

==Help==

{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) : 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x{{=}}1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than $2R$ and each wagon has radius exactly $R$ both wagons must collide at this point.
}}

==References==

[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.

Wagon collision

2017-07-25T02:18:17Z

Oscarlevin: /* Help */

This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

==Puzzle==

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

==Help==

{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) \suchthat 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x{{=}}1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.
}}

==References==

[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.

Wagon collision

2017-07-25T02:17:41Z

Oscarlevin: /* Help */

This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

==Puzzle==

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

==Help==

{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) \suchthat 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.
}}

==References==

[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.

Wagon collision

2017-07-25T02:17:19Z

Oscarlevin: /* Help */

This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

==Puzzle==

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

==Help==

{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) {{|}} 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.
}}

==References==

[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.

Wagon collision

2017-07-25T02:16:53Z

Oscarlevin: /* Help */

This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

==Puzzle==

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

==Help==

{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) : 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.
}}

==References==

[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.

Wagon collision

2017-07-25T02:16:38Z

Oscarlevin: /* Help */

This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

==Puzzle==

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

==Help==

{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) \| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.
}}

==References==

[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.

Wagon collision

2017-07-25T02:16:13Z

Oscarlevin: /* Help */

This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

==Puzzle==

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

==Help==

{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y)| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.
}}

==References==

[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.

Wagon collision

2017-07-25T02:15:45Z

Oscarlevin: /* Help */

This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov.

==Puzzle==

Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?

==Help==

{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I = \{(x,y)| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y = 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.
}}

==References==

[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.

One-way roads

2017-07-25T02:11:30Z

Oscarlevin: /* Puzzle */

Found this one on the SUNY Stony Brook Math Problem of the Month archives.

==Puzzle==

A county has <m>n > 4</m> cities. Is it possible to connect some pairs of cities by one-way roads so that one can travel from every city to every other city using only one or two roads? If so, how? Note that for every pair of cities A and B, only one road connecting A with B is allowed; this road leads from A to B or from B to A, but not both ways.

[[Category: Graph theory]]
[[Category: New]]
[[Category: Needs solution]]

One-way roads

2017-07-25T02:10:58Z

Oscarlevin: /* Puzzle */

Found this one on the SUNY Stony Brook Math Problem of the Month archives.

==Puzzle==

A county has <math>n > 4</math> cities. Is it possible to connect some pairs of cities by one-way roads so that one can travel from every city to every other city using only one or two roads? If so, how? Note that for every pair of cities A and B, only one road connecting A with B is allowed; this road leads from A to B or from B to A, but not both ways.

[[Category: Graph theory]]
[[Category: New]]
[[Category: Needs solution]]

Draft:Bpow101

2016-04-03T21:23:03Z

Oscarlevin: Created page with "Suppose you have 19 hexagons arranged in roughly a 3x3x3 hexagon (so there is a center hexagon, surrounded by 6, surrounded by 12, just like in Settlers of Catan). Place the..."

Suppose you have 19 hexagons arranged in roughly a 3x3x3 hexagon (so there is a center hexagon, surrounded by 6, surrounded by 12, just like in Settlers of Catan). Place the numbers 1 through 19 into these 19 hexes (once each) so that the sum along any diagonal is the same. Note that in each of the three directions, there are two 3-hex diagonals, two 4-hex diagonals and one 5-hex diagonal.

[[Category:MCP]]

{{Bpow}}

Condominium

2015-04-05T01:15:19Z

Oscarlevin:

==Puzzle==

In a certain condominium community, 2/3 of all the women are married (to men) and 3/5 of all the men are married (to women). What fraction of the entire condominium community is married?

{{Solution| The fraction of married women (M) to unmarried women (U) and the fraction of married men (M) to unmarried men (U) can be visualized as follows.

MMMMMMUUU (Women)

MMMMMMUUUU (Men)

We know that the number of women who are married equals the number of men who are married. So the women’s M region and the men’s M region must be the same size. By using the same units in both the women’s diagram and the men’s diagram, we see that the two M regions are six units long each. This represents a total of 12 units out of a total possible of 19 units (9 from the women plus 10 from the men). So 12/19 of the condominium community is married.}}

[[Category: Algebra]]

File:Trolls.png

2015-03-31T16:21:55Z

Oscarlevin:

Too many trolls

2015-03-31T16:20:42Z

Oscarlevin: Created page with "200px This is a logic puzzle I made up for UNC's Math Challenge Problem ==Puzzle== While walking through a fictional forest, you come upon a large..."

[[File:trolls.png|right|200px]]

This is a logic puzzle I made up for UNC's Math Challenge Problem

==Puzzle==

While walking through a fictional forest, you come upon a large group of trolls. The trolls all look identical, but you know that some of the trolls are knights who always tell the truth, while the rest of the trolls are knaves who always lie. Each troll makes a single statement.

The first troll says, "Hi, I'm Tucker."

The remaining 41 trolls each say, in order, "If the previous troll is a knight, then by the time I'm done speaking you will have heard more lies that truths."

Is the first troll's name really Tucker? And which of the remaining trolls are knight and which are knaves?

==Help==

{{Hint| Suppose the first troll is a knave. What does this tell you about the second troll's statement? Conclude that the third troll is impossible.}}

{{Answer| Troll 1 is named Tucker. Trolls will then alternate between being knaves and knights.}}

{{Solution| Suppose that the first troll is a knave. This makes the second troll's statement true, since the hypothesis of his implication is false. Thus troll 2 is a knight. Troll 3 cannot be a knight for if he were, then it would follow that by the time he was done speaking, you would have heard more lies than truths, but you would have heard two truths and one lie. But also the Troll 3 cannot be a knave, for this would make his hypothesis true and conclusion false, meaning that you had not in fact heard more lies than truths, but you had heard two lies and one truth, a contradiction. Thus it is impossible for the first troll to be a knave.

So the first troll really is a knight (and thus named Tucker). This makes the second troll's statement false (as the previous troll really is a knight and you will not have heard more lies than truths). The third troll will then be telling the truth (the second troll is not a knight). Troll 4 will be lying again, and so on. Every odd-numbered troll will be a knight and every even numbered troll will be a knave.}}

==See also==

*[[Chests of logic]]
*[[Chests of logic 4]]
*[[Two guards]]
*[[Three princesses]]

[[Category: Logic]]
[[Category: Cases]]
[[Category: New]]
[[Category: MCP]]

Draft:Bpow29

2015-01-19T17:04:23Z

Oscarlevin:

A fly sits on the corner of a solid wooden cube. What is the shortest distance it must travel in order to reach the opposite corner of the cube?

{{Bpow}}

Rabbit row

2014-11-16T18:21:01Z

Oscarlevin: Created page with "Here is a puzzle based on a neat graph theory counting problem. ==Puzzle== Eleven white rabbits live in eleven white houses, all in a row. One day, the rabbits get together..."

Here is a puzzle based on a neat graph theory counting problem.

==Puzzle==

Eleven white rabbits live in eleven white houses, all in a row. One day, the rabbits get together and decide that they should spruce up their rabbit row by painting some or all of their houses. They decide that while they don't necessarily need to paint every single house, they will definitely NOT leave any two adjacent houses white. How many choices do they have for which collection of houses to paint?

==Help==

{{Hint | Try solving the pattern for smaller numbers of houses and look for a pattern.}}

{{Answer | There are 233 different collections of houses which could be painted.}}

{{Solution | Perhaps surprisingly, if you start with <m>n</m> houses, the number of collections of houses which could be painted given this restriction is the <m>n+2</m>nd Fibonacci number. To see this, note that with 1 house, there are 2 collections (either paint or don't paint the one house). With 2 houses, there are 3 collections (paint the first, second, or both houses). Now inductively suppose that you want to paint <m>n</m> houses. You could either paint or not paint the first house. If you paint the first house, the remaining <m>n-1</m> houses need to be painted, and we know how to do that. If you don't paint the first house, then you ''must'' paint the second house, and then have your choice of how to paint the remaining <m>n-2</m> houses, which we know how to count.}}

==Variations==

Another group of eleven white rabbits also live in eleven white houses, but these are positioned in a large circle. Again, they want to repaint some or all of the houses, leaving no two adjacent houses white. How many ways can they do this?

Of course, we could also ask these questions and include paint color choices. For example, what if every house would be painted red, white or blue, but we don't want any two adjacent houses to be colored identically. How many choices do the rabbits have?

==Mathematics==

The original puzzle asks for the number of independent sets of the path graph <m>P_{11}</m>. An independent set is a set of vertices in a graph no two of which are adjacent (connected by an edge). The first variation asks for the number of independent sets in a cycle graph. The second variation asks for the number of proper 3-colorings of such graphs.

[[Category: Combinatorics]]
[[Category: Graph theory]]
[[Category: Induction]]
[[Category: Sequences]]

__NOTOC__

Domino circuit

2014-11-10T22:46:50Z

Oscarlevin: Created page with "==Puzzle== A domino consists of two squares, each with some number of dots between 0 and 6 in each square. A standard ''double-six'' set of dominoes contains exactly one domi..."

==Puzzle==

A domino consists of two squares, each with some number of dots between 0 and 6 in
each square. A standard ''double-six'' set of dominoes contains exactly one domino with each possible
pair of numbers on it. Suppose you start laying down a line of dominoes, observing the
rule that two dominoes can touch only if the numbers on the touching squares are equal. After laying down all but the last domino you notice that the two ends happen to have 3 and 5 dots respectively. What does the last domino look like?

[[Category: MCP]]

Penguin lineups

2014-11-10T22:39:21Z

Oscarlevin: Created page with "Here is a counting problem based on Gilbreath permutations. ==Puzzle== You have 10 penguins, each a different height. You want to take a photo of the penguins in a single s..."

Here is a counting problem based on Gilbreath permutations.

==Puzzle==

You have 10 penguins, each a different height. You want to take a photo of the penguins in a single straight line. First though, you select some number of the penguins to be looking slightly to the right, and the others to be pointing slightly to the left. For fun, you decide that all the right-looking penguins should be increasing in height while all the left-looking penguins should be decreasing in height, as you move from right to left. How many different such arrangements are possible?

[[Category: Combinatorics]]

Draft:Bpow65

2014-11-10T22:18:08Z

Oscarlevin: Created page with "Given any nine points in a unit square, must there be among the triangles having vertices on the given points at least one whose area is no more than 1/8? {{Bpow}} Categor..."

Given any nine points in a unit square, must there be among the triangles having vertices on the given points at least one whose area is no more than 1/8?

{{Bpow}}
[[Category: MCP]]

Draft:Bpow64

2014-11-10T22:15:38Z

Oscarlevin: Created page with "A field in the shape of a right triangle is to be subdivided into two fields of equal size by building a straight fence between two sides of the field. If the lengths of the..."

A field in the shape of a right triangle is to be subdivided into two fields of equal size by building a straight fence between two sides of the field. If the lengths of the sides of the field are 300 and 400 feet long, respectively, with the hypotenuse being 500 feet long, what is the shortest fence that will do the job and where should the fence be built?

For the more adventurous: Solve the same problem for a right triangular field of arbitrary dimensions.

For the thrill seeker: Solve the same problem for a triangular field of arbitrary shape.

{{Bpow}}

Draft:Bpow63

2014-11-10T22:14:10Z

Oscarlevin: Created page with "A straight bar of metal, initially 800 feet long, expands 8 inches in length. The ends are fixed so that the bar becomes distorted into the shape of an arc of a circle, for wh..."

A straight bar of metal, initially 800 feet long, expands 8 inches in length. The ends are fixed so that the bar becomes distorted into the shape of an arc of a circle, for which the original bar would now be a chord. What is the approximate height above the center of the original bar of this new distorted bar?

Note: An accurate numerical solution to this problem is acceptable.

{{Bpow}}

Draft:Bpow62

2014-11-10T22:12:45Z

Oscarlevin: Created page with "A rectangular block of wood has for its three dimensions a different odd prime number of inches. Its volume and (total) surface area are, respectively, a three digit number an..."

A rectangular block of wood has for its three dimensions a different odd prime number of inches. Its volume and (total) surface area are, respectively, a three digit number and a four digit number. What are the dimensions of the block?

{{Bpow}}

Draft:Bpow61

2014-11-10T22:06:30Z

Oscarlevin:

Two ferry boats sail back and forth across a river, each traveling at a constant speed, and turning back without any loss of time. They leave opposite shores at the same instant, pass for the first time 700 feet from one shore, continue on their way to the banks, return and pass for the second time 400 feet from the opposite shore. What is the width of the river?

Can you solve this without writing down any equations?

{{Bpow}}

[[Category:MCP]]

Draft:Bpow61

2014-11-10T22:05:26Z

Oscarlevin: Created page with "Two ferry boats sail back and forth across a river, each traveling at a constant speed, and turning back without any loss of time. They leave opposite shores at the same insta..."

Draft:Bpow60

2014-11-10T21:59:39Z

Oscarlevin:

Find an ordered list of positive integers, <m>a_1, a_2,\ldots</m> with the fewest possible integers, satisfying all the following properties:

# Every positive integer is the sum of numbers from the list,
# no number on the list appears more than once in any one sum, and
# no two consecutive numbers on the list, that is, <m>a_k</m>, <m>a_{k+1}</m>, appear in any one sum.

(Note that an integer is considered to be the sum of one number on the list if it is actually on the list. Also, for the terminally picky, "fewest possible" refers to inclusion, not to cardinality.)

In symbols, for any positive integer <m>n</m>

:<m>n = \sum b_i a_i</m>

where the sum runs over all elements of the list, <m>b_i = 0</m> or 1, and <m>b_i b_{i+1} = 0</m>.

{{Bpow}}

Draft:Bpow60

2014-11-10T21:59:10Z

Oscarlevin: Created page with "Find an ordered list of positive integers, <m>a_1, a_2,\ldots</m> with the fewest possible integers, satisfying all the following properties: # Every positive integer is the..."

Find an ordered list of positive integers, <m>a_1, a_2,\ldots</m> with the fewest possible integers, satisfying all the following properties:

# Every positive integer is the sum of numbers from the list,
# no number on the list appears more than once in any one sum, and
# no two consecutive numbers on the list, that is, <m>a_k</m>, <m>a_{k+1}, appear in any one sum.

(Note that an integer is considered to be the sum of one number on the list if it is actually on the list. Also, for the terminally picky, "fewest possible" refers to inclusion, not to cardinality.)

In symbols, for any positive integer <m>n</m>

:<m>n = \sum b_i a_i</m>

where the sum runs over all elements of the list, <m>b_i = 0</m> or 1, and <m>b_i b_{i+1} = 0</m>.

{{Bpow}}

Draft:Bpow59

2014-11-10T21:53:13Z

Oscarlevin:

Describe as explicitly as you can all cubic polynomials with integer coefficients having

# three distinct real roots,
# local maximum and minimum values at integers, and
# point of inflection at an integer.

An example of such a polynomial is <m>2x^3 - 18x^2 + 30x + 23</m>.

{{Bpow}}

Draft:Bpow59

2014-11-10T21:50:49Z

Oscarlevin: Created page with " Describe as explicitly as you can all cubic polynomials with integer coefficients having (a) three distinct real roots, (b) local maximum and minimum values at integers, and..."

Describe as explicitly as you can all cubic polynomials with integer coefficients having

(a) three distinct real roots,
(b) local maximum and minimum values at integers, and
(c) point of inflection at an integer.

An example of such a polynomial is <m>2x^3 - 18x^2 + 30x + 23</m>.

{{Bpow}}

Draft:Bpow58

2014-11-10T21:49:04Z

Oscarlevin: Created page with "Consider the graph of <m>f(x) = x^2</m>. Imagine a circle of radius 1 rolling along the x-axis toward the origin until the circle just comes into contact with the graph. Where..."

Consider the graph of <m>f(x) = x^2</m>. Imagine a circle of radius 1 rolling along the x-axis toward the origin until the circle just comes into contact with the graph. Where does the center (a,b) of the circle come to rest?

For the more adventurous, consider the same question for the function <m>f(x) = x^n</m>, for n a positive real number.

For the truly death defying, consider the same question for an arbitrary increasing function having second derivative of constant sign.

{{Bpow}}

Draft:Bpow57

2014-11-10T21:46:03Z

Oscarlevin: Created page with "My dog, Scooter, got a new leash for his birthday. This leash is made of a flexible rubber coil, much like that on the handset of your telephone, which stretches when Scooter..."

My dog, Scooter, got a new leash for his birthday. This leash is made of a flexible rubber coil, much like that on the handset of your telephone, which stretches when Scooter walks away from me and shortens as he walks toward me. The result being that the leash goes directly from my hand to his collar in a straight line without dragging on the ground or becoming entangled in his (very active) feet.

Last night Scooter and I went for a walk. We were both walking at a constant rate of 6 feet per second, and he was 8 feet in front of me. As we came to a corner, he turned and continued to walk at the same rate; when I got the corner, I turned and we continued on our walk.

After Scooter has turned the corner, but before I have reached it, is the leash growing, shrinking, or staying the same size? If its length is changing, at what rate is it changing just when Scooter turns the corner and when does it reach its maximum/minimum length?

{{Bpow}}

Draft:Bpow55

2014-11-10T21:43:16Z

Oscarlevin:

Take any two positive integers <m>N</m> and <m>a</m>. Show that <m>N^a</m> is the sum of <m>N</m> consecutive odd integers.

As an easy example, note that

: <m>7^{13} = 96889010407 = 13841287195 + 13841287197 + 13841287199 + 13841287201 + 13841287203 + 13841287205 + 13841287207</m>

{{Bpow}}

Draft:Bpow55

2014-11-10T21:42:03Z

Oscarlevin: Created page with "Take any two positive integers <m>N</m> and <m>a</m>. Show that <m>N^a</m> is the sum of <m>N</m> consecutive odd integers. As an easy example, note that <m>7^{13} = 96889..."

Take any two positive integers <m>N</m> and <m>a</m>. Show that <m>N^a</m> is the sum of <m>N</m> consecutive odd integers.

As an easy example, note that

<m>7^{13} = 96889010407 = 13841287195 + 13841287197 + 13841287199 + 13841287201 + 13841287203 + 13841287205 + 13841287207</m>

{{Bpow}}

Draft:Bpow54

2014-11-10T21:25:00Z

Oscarlevin:

What is the largest figure eight which fits into a right triangle? For the purposes of the problem, the loops of the figure eight are of the same size.

For the more adventurous: What is the answer if the figure eight is allowed to have loops of different sizes?

{{Bpow}}

Draft:Bpow54

2014-11-10T21:24:00Z

Oscarlevin: Created page with " What is the largest figure eight which fits into a right triangle? For the purposes of the problem, the loops of the figure eight are of the same size. For the more adventur..."

Five card draw

2014-11-10T03:16:25Z

Oscarlevin: Created page with "This puzzle was inspired by a magic trick performed by Ricky Jay on the tonight show. ==Puzzle== Take a well shuffled deck of cards and deal off 10, face up, in a single lin..."

This puzzle was inspired by a magic trick performed by Ricky Jay on the tonight show.

==Puzzle==

Take a well shuffled deck of cards and deal off 10, face up, in a single line. You and a friend are going to take turns taking a card from either end of the line. The player with the better five card hand wins. Do you want to go first or second?

{{Hint | With a little planning (and false shuffling) you can make this into more of a magic trick by dealing the cards face down.}}

==See also==

*[[Coin game]]

[[Category: New]]
[[Category: Algorithms]]
[[Category: Combinatorics]]

MediaWiki:Sidebar

2014-11-09T03:18:39Z

Oscarlevin: Undo revision 1337 by Oscarlevin (talk)

Help:Contents

2014-11-09T03:17:23Z

Oscarlevin:

These are pages designed to help in the maintenance of the site. General help on using a wiki can be found [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents here]. If you are looking for help on a specific puzzle, you are on you own.

==Help pages==

{{Special:Allpages/Help:}}