A woman noticed that her age and her child's age are both prime numbers and that the sum of their ages is a perfect square. How unusual she thought. Then she realised that this will occur again in the future - two times in fact (if they both live to a very ripe old age). How old are they?

Average

I set out on a journey with the intention of completing it at a certain average speed. Unfortunately my average speed for half the journey is only half the average speed I want to achieve for the whole journey. At what average speed do I have to complete the second half of the journey to reach my target for the whole journey?

Bee

Two trains are approaching each other on a straight track, each travelling at 15 miles per hour. They are initially 60 miles apart when a bee takes off from the front of one train at 40 miles per hour and flies to the second train. The bee then flies back and forth between the approaching trains until the trains meet. How far does the bee travel?

Big numbers

Multiply together two numbers formed by the digits 1 to 9 each digit used once. How big a number can you get?

Birthday party

How many people need to be at a party so that the probability of there being two people with the same birthday is fifty-fifty?

Boxes

I start with a rectangle and cut a square from each corner in such a way as to maximise the volume of the resultant box formed by folding up the sides. It turns out that the squares I cut out are unit squares (1 inch sides) and the resultant box has a length and width both of which are an integer number of inches. What are the dimensions of the original rectangle?

Cells

There are 100 prisoners in 100 separate locked cells. During the night each of 100 prisoner officers visits the cells. The first officer visits every cell. The second officer visits cells 2,4,6,... etc (every 2nd cell), the third officer visits every third cell, the fourth officer visits every fourth cell and so on until the 100th officer visits the 100th cell. On a visit each officer unlocks the door if it is locked or locks the door if it is unlocked. In the morning, how many prisoners escape and why?

Christmas tree lights

My Christmas tree is 6 feet tall with a base radius of 3 feet. What is the length of Christmas tree lights I need if I want there to be 10 equally spaced turns in the spiral of lights? Is there an easy way to estimate the length required without the complication of working with arc lengths on the surface of a cone? Would your estimate be more accurate with additional turns?

Circles and Areas

Prove: Area of a = Area of b

(Puzzle from @mathsjam)

Circular argument

I have a 1 foot diameter hoop that I use to mark out circular 1 foot diameter plots in the garden. How can I use the same hoop to mark out plots 2 feet in diameter?

Cover the squares?

Can you cover the large figure with rectangles - size as shown? No overlaps inside or outside the figure.

Crazy curves

Except for the circle (which is shown for reference), on the other three curves, except where the curves cross an axis, all the points (x,y) have an interesting property. In fact this property holds for all the curves of this family for any power of x and y above 2. What is so special about the co-ordinates (x,y) of these points and why?

Cross the bridge

Four people have to cross a narrow bridge at night. The bridge can only take a maximum of two people at a time and they must use a torch. They only have one torch. Individually they can cross in 1, 2, 5 and 10 minutes. If two travel together they must go at the speed of the slowest. How do they all get across in 17 minutes?

Cube from sphere

What is the volume of the largest solid cube that can be cut from a solid sphere of volume V?

Efficient containers

Calculate the ratio of the surface areas of a sphere and a cube where the sphere and the cube both contain the same volume. Can you prove that a sphere is the most efficient container? (Prove that a sphere is the container that has the minimum surface area containing any given volume.)

Eggs

I divide up a number of eggs between three people. I give the first person half the eggs and half an egg. From the remainder, I give the second person half the eggs and half an egg. From the remainder, I give the last person half the eggs and half an egg. I have none left. How many eggs did I start with if there is no mess?

Escalator

Two escalators run side by side, one up and one down. Both move at the rate of 60 steps per minute. As I go up I pass 60 people coming down. It's a busy time. I notice every step is occupied by one person for the whole time and nobody is walking up or down. How long was I on the escalator?

Find the centre

Given a circle, find its centre using only pencil, ruler and compass.

Integration

Given that the function INT(x) takes the integer part of a real variable x, what is the integral of INT(x)? Can you evaluate the integral from x=0 to x=5?

Ladder

The diagram shows a ladder slipping down a wall, starting vertical (red) and finishing horizontal (blue) and going through the position shown (grey). What is the path of the mid-point of the ladder? What is the path of any point x up the ladder, where x ranges from 0 to 8 in this example?

Logic - Hats

Two red and two green party hats are placed on four people and these people are arranged as shown - so that A can see B & C, B can see C, but C and D cannot see any of the others. (They all know that there are two red and two green hats.) The first person to correctly call out the colour of the hat on their head wins a prize. Who wins, given that nobody simply guesses?

My watch

Yesterday I decided to use a watch I have not worn for many months.Although the time was correct, the date dial showed 5 when in fact it was September 13 2013.In which month of which year did I last wear this watch?

Mystic Rose

The Mystic Rose is formed by placing equally spaced points around the circumference of a circle and then joining each point to all the others. The example shown is generated from 8 points. If n is the number of points from which the Rose is formed, can you tell me, in terms on n, how many points are formed by the intersection of lines making up the Rose?

Pegs

Which is the best fit? A square peg in a round hole or a round peg in a square hole.

Quadratic roots

The difference between the roots of a quadratic is 2. What can you say about this family of quadratic equations?

Rectangle

Consider a rectangle with length l and width w. Can you find values for l and w such that the numerical value of the perimeter of the rectangle is equal to the numerical value of the area of the rectangle? Can you find solutions where l and w are both integers? How many such solutions are there?

Shadow

Have you ever watched the path of the shadow of somebody walking past a lamppost? What path does the end point of the shadow take? How long is the shadow?

Ship

A container ship is on an enclosed lake which is perfectly calm. One of the on-board containers slips off the ship and sinks. After the lake has settled again to be perfectly calm, is the height of the water higher, lower or the same height as before?

Square root

Draw a line x cms long. Now using ruler and compass only construct a line whose length is the square root of x.

Square window

Given a square window which is, say, 3 feet high and 3 feet wide, make another window with half the area which is still square and is also 3 feet high and 3 feet wide.

Squares

Try this with n=2 and then with n=3. Can you generalise to the full solution?

Throw

Imagine two people standing 10 feet apart – each throwing a ball to the other at the same speed of 10 feet/second, releasing the ball, at the same height, horizontally at the same time. How long before the balls meet and where do they meet?

Now imagine one person is 10 feet above the other (leaning out of a window perhaps!). They throw the ball to each other at 10 feet/second – one upwards and one downwards. How long before the balls meet and where do they meet?

Twenty

10+10=20 and the product of 10x10=100. Can you find a set of positive integers that sum to twenty and give the highest possible product when the integers are multiplied together?

Water and wine

A glass of wine. An equal glass of water. Take one spoonful of water and pour into wine. Mix thoroughly. Take one spoonful of mixture and pour into water. By volume, is there more water in the wine or more wine in the water?

Wine

A glass of wine. An equal glass of water. Take one spoonful of water and pour into wine. Mix thoroughly. Take one spoonful of mixture and pour into water. By volume, is there more water in the wine or more wine in the water?