Which pins must be knocked over to score exactly 100 points? (Hint: There are three!)

Answer: 13, 39, and 48.

96 = 2^5 x 3^1

Q: Without a calculator or the Internet, find the prime factors of 2023.

A: Try prime numbers.
First: 2 the number is not even so 2 is not a factor.
For 3, we have to check the sum of the digits which total to 7 which is not divisible by 3, therefore 2023 is not divisible by 3.
For 5, the number doesn’t end in 5 or 0 so not divisible by 5.
For 7, think of a number close that is divisible by 7 — 2100 is 300 x 7.
2023 is 77 less than 2100. Since 77 is also divisible by 7, 2023 is a multiple of 7. It is 300 x 7 – 11 x 7 which is 289 x 7.
289 happens to be a perfect square as it is 17 x 17.

Since 17 is prime, it cannot be factored any further.

So the prime factors of 2023 are: 7 x 17 x 17.

Q: Suppose you want to cut a circular cake with six straight cuts. What is the maximum number of pieces you can create? They can be different sizes but cannot be moved around.

A: 22 pieces.
The figure below is for 5 cuts with 16 pieces.
Each additional cut creates one more additional piece than the previous cut.
Here is the pattern:

#of Cuts #of Pieces # of Added Pieces
1 2 0
2 4 2
3 7 3
4 11 4
5 16 5
6 22 6

Q: Suppose you want to cut a circular cake with five straight cuts. What is the maximum number of pieces you can create? They can be different sizes but cannot be moved around.

A: 16 pieces…see below for a possible solution