Maths Teaser.

locum-motion

From Yesterday's Sunday Times:

"Willy took a cylindrical cake of radius a whole prime number of millimetres. He coated the curved surface with a layer of chocolate an odd prime number of millimetres thick and let it set. He repeated this a few times, building up layers of the same thickness. Being a silly Willy, he left the cake in the sun and the chocolate melted. So he used it to make nine solid cylinders of chocolate, each of radius 20mm and the same length as the original cake.
How many layers of chocolate did he apply to the cake?"

Here's my answer:

Let:
r = radius of cake {a prime}
R = radius of cake with chocolate layers added {= r + (a whole no x an odd prime)}
l = number of layers
20 = radius of chocolate cylinders
h = height of the cake and of the cylinders

Volume of chocolate = 9 x Pi x 20^2 x h
= 9 x Pi x 400 x h
= 3600 Pi x h.

Volume of Cake + Choc = Volume of Cake + Volume of Choc
=> R^2 x Pi x h = r^2 x pi x h + 3600 x Pi x h
=> R^2 = r^2 + 3600 (divide all elements by Pi x h)
=> R^2 -3600 = r^2

Then, a few minutes sticking primes into a little Excel file I made gives:

61 ^2 = 3721
3721 - 3600 = 121
11 x 11 = 121

So, Willy put 11 layers of chocolate, each 11mm thick, onto a cake of radius 61mm.

Is that right? Was there any easier way of doing it?

hivizman

I agree with your calculation, but not with the interpretation of your result. What your result means is the following:

r = 11, R = 61, so the original cake had a radius of 11mm and the chocolate layers added an extra 50mm to give the iced cake with radius 61mm. Hence Willie must have put 10 layers, each 5mm wide, on the cake.

I did this by rearranging your equation to get R^2-r^2=3600. The left hand side can be factored as (R+r)*(R-r)=3600. Set R+r=F and R-r=f, then the problem reduces to splitting 3600 into two factors F and f such that (F-f)/2, which equals r, is a prime. The only split that works is 3600=72*50, giving r=11 and R=61.

Although I confirmed the result by testing all factorisations of 3600 using Excel, you would get straight to the solution if you spotted that {11, 60, 61} is a Pythagorean triple, so 11^2+60^2=61^2. However, you would need to check that the solution is unique.

locum-motion

hivizman wrote: »

I agree with your calculation, but not with the interpretation of your result. What your result means is the following:

r = 11, R = 61, so the original cake had a radius of 11mm and the chocolate layers added an extra 50mm to give the iced cake with radius 61mm. Hence Willie must have put 10 layers, each 5mm wide, on the cake.

I did this by rearranging your equation to get R^2-r^2=3600. The left hand side can be factored as (R+r)*(R-r)=3600. Set R+r=F and R-r=f, then the problem reduces to splitting 3600 into two factors F and f such that (F-f)/2, which equals r, is a prime. The only split that works is 3600=72*50, giving r=11 and R=61.

Although I confirmed the result by testing all factorisations of 3600 using Excel, you would get straight to the solution if you spotted that {11, 60, 61} is a Pythagorean triple, so 11^2+60^2=61^2. However, you would need to check that the solution is unique.

Sorry, 61 is the radius of cake and choc together. My bad.