
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.
 Suppose the whole plane is coloured, with three colours (say
Red, Blue and Yellow). All three colours must be used.
 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?)
 Is it always possible to draw a triangle,
each corner of a different colour, such that the perimeter is
of a given length?
 Is it always possible to draw a triangle,
each corner of a different colour, such that the area is
of a given value?
 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?
 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?
 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?
 Into how many pieces can you cut a pizza using n straight cuts?

 3^{2}+4^{2} = 5^{2}. 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.
 5^{2}+12^{2} = 13^{2}. Design a jigsaw puzzle based on this equation.
 What about 7^{2}+24^{2} = 25^{2}?
 What about the general case where a^{2}+b^{2} = c^{2}, for a, b and
c whole numbers?
 What if your pieces must have only right angles?
 What about equations like 5^{2}+5^{2} = 1^{2}+7^{2}? Design a jigsaw puzzle
that can be assembled either into two 5cm squares, or into
a 1cm square and a 7cm square. What about 1^{2}+8^{2} = 4^{2}+7^{2}?
 What about 3 dimensions? 3^{3}+4^{3}+5^{3} = 6^{3}. 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?
 What about 1^{3}+12^{3} = 9^{3}+10^{3}? or 2^{3}+16^{3} = 9^{3}+15^{3}?
 Everyone learns about rulerandcompass 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 twosided rule and no compass? At least, I've
been assured that is the case...
 How can you bisect an angle using a twosided ruler?
 How can you construct a 90 degree angle? A 60 degree angle?
 From these, construct a square, and a regular hexagon.
 How do you bisect a line segment?
 Can you construct a regular pentagon?

 The 19th of September, 1991 was a palindromic calendar date 
19991. 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, 11011, 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?
 In fact, the 19th of September, 1991 would also be palindromic
if you wrote the year in full: 1991991. Let's call a date
'superpalindromic' 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 superpalindromic. Was this
the last superpalindromic date of the 20th century? Was it the only one?
Is every superpalindromic date also palindromic? How many superpalindromic
dates have there been in the past 1000 years? What about in the next 1000 years?
If you wanted to be born on a superpalindromic date, what century should you have
been born in to have the best chance? Can a year ever contain two
superpalindromic dates?
 Here's the American version of the first puzzle:
The 9th of August, 1998 was a palindromic calendar date 
8998. 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, 11011, 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?

 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?
 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?
 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?
 What if there are 6 towns? 7? Find a general formula.
 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).
 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?
 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?
 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?
