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Games and Puzzles

Mathematical games puzzles and problems.

Please note that many of the puzzles below are taken or adapted from other sources and are not original. We are continuing to expand our selection of games and puzzles, so if you have any puzzles we could include, please send them together with your solutions and comments to webmaster@ima.org.uk.

Dice Rolls
Games and PuzzlesA standard dice is tossed until the total thrown exceeds 12. What is the most likely final total? For example, if it came up 3, 6, 1, 5 then the total would be 15.

Solution
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The Mirascope
MirascopeA Mirascope uses parabolic shaped mirrors to create an illusion. It is basically comprised of two almost identical mirrors, each in the form of a paraboloid of revolution, facing each other, one on top of the other. It looks rather like a flying saucer which has an object floating on top in mid-air! The floating object is actually the incorporeal projection of a physical object placed inside. From certain viewing angles you can only see the floating image and it is tempting to try to grasp it. Of course, your fingers grasp nothing! The photograph is taken from an angle which allows you to see both the physical object inside (in this case a model of the Tardis!) and a rather distorted floating image.

How is the image generated? Follow the link to a dynamic simulation of a cross-section of a Mirascope which might help you figure it out!

Balance
Games and PuzzlesYou are given a box of nine balls. Eight are of the correct weight but one is too heavy. The only way you can weigh the balls is with a balance; i.e. you can weigh one set of balls against another. What is the minimum number of weighing operations you need to be sure of identifying the faulty ball?


Solution

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Let t be the (positive) square root of 2.
What is the 62nd digit after the decimal point of (1+t) to the power 2012?

Even if your calculator were able to help with this, you don't need it!

Hint: It would be nice to work with an expression without the awkward square root of 2.Let t be the (positive) square root of 2. What is the 62nd digit after the decimal point of (1+t) to the
power 2012?

Solution 
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Card Game
Card GameKen and Barbie are gambling on a card game with £96 in the pot. A standard pack of cards is shuffled and the cards are turned over two at a time. If the cards consist of one red and one black they are discarded; if they are both red Ken gets them and if they are both black Barbie gets them. At the end of the game the player with more cards wins.

They have reached the point where four cards are remaining when the game has to be abandoned. At this point Ken has two cards more than Barbie. They argue about how to divide the stake.

Ken argues, “The Law of Averages says that over the last two rounds each of us will do equally well, so I will win the game and should collect all the money.”

Barbie argues, “When two cards are turned over there is a 0.5 probability that there is one of each colour, 0.25 that it is two reds and 0.25 that it is two blacks. There is therefore a 1 in 16 chance that I will win the game by getting four cards from the last two rounds, so I should get 1/16 of the stake. It should be divided £90 to Ken and £6 to me.”

Who is right? Why? How should the money be divided most fairly?

Solution 
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Number sequence
Can you spot the next number in the sequence 1, 11, 121 ,1331, 14641, ?, ...
Submitted by Dr John D. Mahony CMath FIMA

Solution 
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Frequency of numbers in Pascal's triangle
Puzzles and Games In Pascal’s triangle the number 1 appears infinitely many times. All other numbers will appear a finite amount of times*.

The number 2 appears just once.
The number 3 appears twice.

What is the first number that appears exactly 3 times?
What is the first number that appears exactly 4 times?

Can you find a number that appears exactly 5 times?
Can you find a number that appears exactly 6 times?
Can you find a number that appears more than 6 times?

*Can you prove this? 

Solution 
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Circle Square
Puzzles and GamesThe diagram shows four circles of radius 1/4 in a unit square

A circle can fit in the gap in the middle.

What is the radius of the circle in the middle? What proportion of the original square is covered by the circle in the middle?

Puzzles and GamesIf you placed eight spheres of radius 1/4 in a unit cube a sphere can fit in the middle.What is the radius of the sphere in the middle? What proportion of the original cube is covered by the sphere in the middle?

What happens in 4 dimensions?

And in higher dimensions how big does the diameter of the hypersphere in the gap get and what proportion of the hypercube is covered?

Solution 
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Not all triangles are perfect, but...
The first two perfect numbers are 6 and 28

A number is perfect if it is equal to the sum of its factors other than itself: e.g. 6 = 1 + 2 + 3 and 28 = 1+ 2 + 4 + 7 + 14.

Games and Puzzles

The first four triangular numbers are 1, 3, 6, 10. Both 6 and 28 are triangular numbers. Are all perfect numbers triangular?

Games and Puzzles

The first four hexagonal numbers are 1, 6, 15, 28. Both 6 and 28 are hexagonal numbers. Are all perfect numbers hexagonal?

Solution 
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The IMA would like to thank Mathematics in Education and Industry (MEI) for supplying the three puzzles above. If you enjoyed these, more puzzles can be found on the MEI website by following the links below.

MEI Maths Item of the Month Archive 2010

MEI Maths Item of the Month Archive 2009

MEI Maths Item of the Month Archive 2008

MEI Maths Item of the Month Archive 2007

MEI Maths Item of the Month Archive 2006

Bee-line 1
BeeA freight train 200m long travels along a straight track at constant speed. In the time it takes the train to move a distance equal to its own length, a bee, initially sitting at the rear of the train, flies forward to the front of the train, turns instantly and returns to the rear, flying at constant speed all the while. What is the total distance travelled by the bee?

Solution 
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Bee-line 2
A bee is sitting at a distance, d, from a man. At time t=0 the bee pursues the man at a constant acceleration of 1. The man runs in a straight line away from the bee with an acceleration of 0.3t. What is the largest initial separation, d, for which the man can’t quite escape the bee?

Solution 
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Boxes
Puzzles and Games
Play against the computer by clicking on two dots to create a single vertical or horizontal line.

The player who draws the fourth and final line that completes a box wins the box and draws another line. You always draw another line after you complete a box so it’s possible to draw several lines during one turn.The game continues until every possible box has been made and there are no more dots to be joined.

To win you must have completed the most boxes.

Chicken Nuggets
puzzlesAt a fast-food take-away you can order chicken nuggets in boxes containing six, nine or twenty nuggets. What is the largest number of nuggets you cannot buy with these boxes?


Solution
 
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Circle Regions
Put n points on the circumference of a circle. Join each point to every other one with a straight line. This cuts the circle into a number of regions. What is the maximum number of regions you can generate from n points?

For example, with two points you just get two regions, with three points you get four regions, with four points you get eight regions.


Circle regions puzzle

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Circular Billiards
puzzlesImagine a circular billiard table of unit radius, with centre at (x,y ) coordinates (0,0). It has two billiard balls on it. Ball A is at coordinates (-0.8, -0.1); ball B at (0.6, 0.4). What are the coordinates of the point(s) on the cushion to which one must hit ball A so that it bounces off the cushion once and then hits ball B? Explore the possibilities.  

 

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Leapfrog 
Bicycle wheelAdam and Brian decide to cycle from Adam’s house to Brian’s house 8 km away. Unfortunately Brian has left his bike at home! They decide to share Adam’s bike, each alternately riding and walking. They agree that the rider will get off the bike when he is a distance of h km ahead of the walker and will continue on foot himself. Once the walker reaches the bike he will start to ride it, and continue riding until he is h km past the other, when he, in turn, will leave the bike and continue again on foot. This process will continue until they reach Brian’s house. They set out together with Adam riding and Brian walking. Both of them walk at a uniform speed of 4 km/hr and ride at a uniform speed of 16 km/hr. What is the minimum time for the journey, as measured by the last one to reach the destination? If the bike changes hands twice during the journey what is the distance h?

(Problem prompted by the Leapfrog chapter in the book Number Crunching by Paul J. Nahin, Princeton University Press, 2011). 

Solution 
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Peg 
SolitaireTry to clear the board by jumping pegs over other pegs. The "jumped" peg will be removed from the field. If there's more then one move available, just choose which move you would like to do. You win by getting it down to just one peg (The best solution is with this peg in the centre).

Radioactive Dice
puzzles
Question: What is the half-life of radioactive dice?
No, really! It’s a serious question (well, half-serious, anyway!)
Definitions: By definition, a radioactive die is one that spontaneously disintegrates if it’s thrown six uppermost. And the half-life is the length of time you have to keep throwing lots of radioactive dice until you are left with exactly half the number you started with (ok, I know you don’t actually have any to start with – use your imagination!). “Time” here is measured in number of throws, of course. 

Methods 1 and 2 
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Snow Plough

puzzlesOne day it started snowing steadily. A snow-plough started out at noon, going two miles in the first hour and one mile in the second hour. What time did it start snowing?

(From: Agnew, Ralph P. Differential Equations. McGraw-Hill. 1960)

Solution 
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The Sultan's Daughters
puzzlesA sultan has 100 daughters. A commoner may be given a chance to marry one of the daughters, but he must first pass a test. He will be presented with the daughters one at a time. As each one comes before him she will tell him the size of her dowry, and he must then decide whether to accept or reject her (he is not allowed to return to a previously rejected daughter). However, the sultan will only allow the marriage to take place if the commoner chooses the daughter with the highest dowry. If he gets it wrong he will be executed! The commoner knows nothing about the distribution of dowries. What strategy should he adopt?

Solution 
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RELATED INFORMATION

Maths Arcade The first Maths Arcade was set up at the University of Greenwich in September 2010. As a result of dissemination of this in Greenwich, Birmingham and Exeter other Arcades have been initiated elsewhere since. More information »

Other sources of puzzles include:

MangaHigh a games-based maths teaching resource. MangaHigh »

Plus magazine Puzzles »

Project Gutenberg 'Amusements in Mathematics' by Henry Ernest Dudeney a collection of mathematical puzzles. 'Amusements in Mathematics' »

Puzzles.com includes a wide variety of mathematical and other puzzles. Puzzles.com »

Sam Loyd.com a collection of Sam Loyd puzzles and classroom resources. Sam Loyd.com »