Earlier today I set you the following three puzzles:
1.You have a square cake, and four friends. How do you divide the cake into five slices of equal size? Each slice must be slice-like, meaning that the knife cuts vertically and the tip of each slice is at the centre of the cake. You have no ruler or tape measure, but you can use the horizontal grid here.
SOLUTION
The solution is for each slice to have the same amount of the perimeter of the cake. (It’s essentially the same answer as dividing a circular cake into five slices). Since the perimeter of the cake is 20 units (as marked by the grid ), then each of the five slices must have 4 units of edge. So, choose a point on the perimeter and then mark all the other points four units along:
When you slice from the centre to each point you are left with five equally sized slices. You may have been trying to find five slices that had the same shape - but the question did not ask for that. The slices look different, but contain the same amount of cake.
We know that the slices are of equal size because the area of each slice is either a triangle, or the combination of two triangles (as shown below). The area of a triangle is half the base times the height. The triangles that make up the slices all have the same height, which is the perpendicular distance from the perimeter to the centre (in this case 2.5units). If the slice is a single triangle the base length is 4, and if the slice is two triangles, the two base lengths add up to 4. So the area of all slices is the same.
(If you don’t believe me we can do the calculations here: the triangle slices have area 1/2 x 4 x 2.5 = 5. The area of the bottom left and top right slice is (1/2 x 1 x 2.5) + (1/2 x 3 x 2.5) = 5. The area of the bottom right slice is 2 x 1/2 x 2 x 2.5 = 5.)
In fact, the solution works for every possible whole number of slices of cake. If you want to slice a cake into 7 or 9 or n slices, divide the perimeter of the square cake into 7 or 9 or n equal lengths.
2. You have a rectangular cake, and two friends. One of your friends (the annoying one) just cut herself a rectangular slice, as below. Show how to divide what’s left into two portions of equal size.
SOLUTION
The piece of insight needed to solve this one is to realise that any straight line through the centre of a rectangle divides the rectangle into two parts of equal area.
Consider the cake before the friend ate the rectangular slice. Any slice through the centre of the cake will divide it into two equal portions. Now consider what happens after the friend ate the slice. If the cake is cut so that it goes through both the centre of the cake and the centre of the slice, as below, the cut will again divide the cake into two equal portions. This is because the gap left by the eaten slice is also split in two, meaning that the area of each of the two equal portions by will be reduced by the same amount, and thus remain of equal size. Although, of course, they do not have the same shape, and one of the portions is made of two pieces.
3. You have a circular cake. You sprinkle exactly 100 hundreds and thousands on the cake. These are mathematical hundreds and thousands, meaning that each one is a single point with no length or width. Show how it is always possible to slice the cake into two with a single cut of the knife so that each slice has exactly 50 hundreds and thousands on it. You can assume that the surface of the cake is perfectly horizontal.
SOLUTION
The solution is theoretical - it gives you a method to find the slice, although in practice it will be very difficult to do!
(I should have clarified that no single hundred and thousand sits on another hundred and thousand. And also, for non-Brits, a hundred and thousand is a sprinkle.)
The first step is to imagine every single line that goes through any two of the hundreds and thousands. The next step is to select any point on the same horizontal plane as the hundreds and thousands that lies outside the cake and not on any of these lines.
Draw a line through this point that just misses the cake, as below, and then rotate this line clockwise around the point like the hand of a clock. The line will hit the cake, and then slowly as you move it across the cake it will cross a single hundred and thousand, and then two, etc, eventually crossing 50 of them. At this point the line is where you should cut the cake. Since you chose a point not on the same line as any two hundreds and thousands, the rotating line will never cross two hundreds and thousands at the same time.
I hope you enjoyed the puzzles. Now I’m off to celebrate my birthday.
I set a puzzle here every two weeks on a Monday. If you would like to suggest a puzzle email me.
My latest book Can You Solve My Problems? A Casebook of Ingenious, Perplexing and Totally Satisfying Puzzles is just out. It contains my favourite puzzles from the last 2000 years, along with historical, biographical and mathematical background. Simon Singh reviewed it here. Available from the Guardian Bookshop and other retailers.
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