 # Puzzles!

A collection of my favourite maths puzzles - some quite hard, some easier, some suitable for a school maths project, some suitable for a research paper... I'll leave it to you to figure out which are which... but if you're looking for math puzzles for kids, I suggest you check out my math games for kids website.

1. Suppose the whole plane is coloured, with three colours (say Red, Blue and Yellow). All three colours must be used.

1. Is it always possible to draw a triangle, each corner of a different colour, such that one of the sides is of a given length? (say 1cm?)
2. Is it always possible to draw a triangle, each corner of a different colour, such that the perimeter is of a given length?
3. Is it always possible to draw a triangle, each corner of a different colour, such that the area is of a given value?
4. Is it always possible to draw a triangle, each corner of a different colour, such that one of the angles is of a given value?

2. Imagine a set of line segments joined end to end in the plane, not crossing each other. If the line segments are stiff rods, pivoted at the corners, is it always possible always to ünbend" the corners to form one single long line segment, without ever any of the line segments crossing each other?
3. Suppose there are n blue points and n red points in the plane, with no three points collinear. Is it always possible to join each red point to a single blue point with a straight line segment, so that no two line segments cross each other?
4. Into how many pieces can you cut a pizza using n straight cuts?
1. 32+42 = 52. Design a jigsaw puzzle with as few pieces as possible which can be assembled either into a 3cm square and a 4cm square, or into a 5cm square.
2. 52+122 = 132. Design a jigsaw puzzle based on this equation.
3. What about 72+242 = 252?
4. What about the general case where a2+b2 = c2, for a, b and c whole numbers?
5. What if your pieces must have only right angles?
6. What about equations like 52+52 = 12+72? Design a jigsaw puzzle that can be assembled either into two 5cm squares, or into a 1cm square and a 7cm square. What about 12+82 = 42+72?
7. What about 3 dimensions? 33+43+53 = 63. Can you design a 3 dimensional jigsaw puzzle which can be assembled either into 3 cubes (of sizes 3cm, 4cm and 5cm) or into a single cube of size 6cm?
8. What about 13+123 = 93+103? or 23+163 = 93+153?

5. Everyone learns about ruler-and-compass constructions, how to bisect an angle, or construct a right angle and so on. But did you know that all these are also possible with a two-sided rule and no compass? At least, I've been assured that is the case...

1. How can you bisect an angle using a two-sided ruler?
2. How can you construct a 90 degree angle? A 60 degree angle?
3. From these, construct a square, and a regular hexagon.
4. How do you bisect a line segment?
5. Can you construct a regular pentagon?

1. The 19th of September, 1991 was a palindromic calendar date - 19-9-91. The digits are the same if you read the date in reverse. At least, this is the case if you put the day before the month (which is the general standard outside of America). The 1st of October 2011 will be special for the same reason, the digits of the date, 1-10-11, read the same backwards as forwards. When is the next such date? When was the last one? How many have there been since you were born?
2. In fact, the 19th of September, 1991 would also be palindromic if you wrote the year in full: 19-9-1991. Let's call a date 'super-palindromic' if the digits read the same backwards and forwards when you write the year in full. Then, the 19th of September 1991 was fortunate to be both palindromic and super-palindromic. Was this the last super-palindromic date of the 20th century? Was it the only one? Is every super-palindromic date also palindromic? How many super-palindromic dates have there been in the past 1000 years? What about in the next 1000 years? If you wanted to be born on a super-palindromic date, what century should you have been born in to have the best chance? Can a year ever contain two super-palindromic dates?
3. Here's the American version of the first puzzle: The 9th of August, 1998 was a palindromic calendar date - 8-9-98. The digits are the same if you read the date in reverse. The 10th of January 2011 will be special for the same reason, the digits of the date, 1-10-11, read the same backwards as forwards. When is the next such date? When was the last one? How many have there been since you were born?

1. Suppose there are three towns, A, B, C. From anywhere in the country, you can rank the three towns in order of how far away they are. Then you can colour the map according to which order prevails: for example, if ABC was blue, ACB was green BAC was red an so on, you would get a nice pattern of 6 colours on the map, since there are 6 ways to arrange the towns. But wait... do you always get 6 colours on the map?
2. Suppose now there are 4 towns, A,B,C,D. Since there are 24 ways to arrange the towns in order, I will buy a box of 24 coloured pencils. Then, I will colours the map according to the ranking of the towns in order of how far away they are from each point on the map. Do I use all 24 colours, or do I need less than that, in fact?
3. Suppose now there are 5 towns? There are 120 different arrangements of the letters A,B,C,D and E, but how many colours do I need to colour my map?
4. What if there are 6 towns? 7? Find a general formula.
5. Now generalize this problem to 3 dimensions. Or 4 dimensions. Or, if your brain is tired, generalize it to 1 dimension (so that all the towns are along a stretch of very straight highway).

6. I have a number of keys on a round keyring. If there are 2 keys, I need two colours to distinguish them. If there are 3, I need 3 colours, but if there are 4, how can I distinguish all 4 keys using only 3 colours? How many colours will I need for 100 keys?
7. Candles are sold in packs of 24. If John celebrates his birthday every year from the day he is born, how often will he have used an exact number of packs of candles? (For example, if candles were sold in packs of 10, after his 4th birthday he would have used exactly 1 pack (1+2+3+4 = 10), and after his 15th birthday he would have used exactly 12 packs (1+2+3+4+...+14+15 = 120), but after every birthday in between he still has some spare candles). What if candles are sold in packs of n?
8. How many different group photos can I take at a gathering of n people? If there is 1 person, A, I can only take 2 photos, one of A, and one of the wall (hey! some walls are very interesting! probably...) If there are two, say A and B, I can take 5 photos - A on B's right, B on A's right, A solo or B solo, or the wall. If there are three people, I can take 16 photos, ABC, ACB, BAC, BCA, CAB, CBA, then AB, BA, AC, CA, BC, CB, then three solo shots and the wall. How many photos can I take if there are 4 people? 5? 6? Can you find a formula for if there are n people?