2018 NSW: Maths Puzzle 4

4th puzzle by Dr Harry Wiggins

It is Nxolo's birthday. She has one big cake and wants to divide it into the many pieces. She knows the number of people at her birthday party is either 20 or 18. What is the least number of pieces, she has to divide her cake so that in either case, she will be able to give everybody the same amount of cake? And how can she achieve this? (Hint: Fractions!)

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8/1/2018 2:12 PM
180 pieces

Let the minimum number of pieces be x. We know that x/20 is an integer, and so is x/18. This means that 18 and 20 are factors of x. LCM(18, 20) = 180, so therefore the minimum number of pieces needed is 180.

She can achieve this by cutting 180 equally sized pieces using a protractor or something
8/1/2018 2:18 PM
Cut the cake into quarters (easy), then trisect each quarter (harder), then trisect each sector again (harder), then split each sector into 5 equal parts (probably not humanly possible, but this is maths so...)
8/1/2018 2:18 PM
Alternatively, take a protractor and cut the cake every 2 degrees (but that's kind of cheating)
8/1/2018 2:44 PM
Does the cake have to be finished, or only be sectioned so that everyone gets the same amount of cake? If it has to be finished then the answer is 180, as that is the LCM of 18 and 20, but if it only needs to be sectioned so that everyone gets an equal amount of cake, then you can cut the cake into 20 equal pieces and give everyone one piece.
Sachin Reddy Northwood School
8/1/2018 3:03 PM
Yes, in either case the WHOLE CAKE must be shared. 180 pieces work, but you can do it with less than 180 pieces. Can you do it with 36 pieces?

(Hint: All the pieces does not have to be the same size. )
8/1/2018 3:05 PM
It does not have to be 180 equal pieces but 36, as you can first divide the cake into 18 pieces, then divide it into 20 pieces in the same way. The number of pieces of cake Seperating the cake into two halves, there will be 8 cuts one one side, to divide it into 9 pieces, and 9 further cuts from the next step, to make 17 cuts per half. Thus there are 18 pieces per half, and 36 pieces of cake.
8/1/2018 4:35 PM
So basically yes, 36 pieces is enough. This is because she can just cut 9/10 of the cake into 18 pieces and then the remaining 1/10 part into 18 pieces.

When there are 18 people, she just needs to give each person a big slice and a small slice and then everyone has 1/18 of the cake

When there are 20 people, she gives the first 18 people a big slice each, and then the remaining 2 people each 9 small slice. This way, everyone has 1/20 of the cake
8/4/2018 5:29 AM
Well done Andi for explaining nicely the size of the 36 pieces. Of course a little proof is needed to show why if you use less than 36 pieces it is impossible.