We’re working with the first nine prime numbers (not counting 1), so that’s 2, 3, 5, 7, 11, 13, 17, 19, and 23.

Step 1: The first clue is “Both main diagonals sum to the same prime number, a number appearing in this puzzle.”

This means that A1+B2+C3 is equal to A3+B2+C1. Since they overlap at B2, A1+C3 must also equal A3+C1. The largest number in this puzzle is 23, so that must be the total of both diagonals. No numbers greater than 7 can be in B2 (e.g. 23-11 = 12, which is impossible to find two sums for). 23-7 = 16, which is also the sum of 3+13 and 5+11. Therefore, B2 = 7.

Step 2: The clue in B2 says “The bottom row is the only one that sums to a prime number”.

We know from Step 1 that the corners are either 3, 13, 5, or 11. This leaves only 17, 19 and 23 as options for B3, A2, and C2. Since we know that the middle row cannot equal a prime number, 19 and 23 must be in either A2 or C2. That leaves 17 for B3.

Step 3: The clue for B3 says “The right most column is the only one that sums to a prime number”.

We know that our options for C1 and C3 are 3, 13, 5 or 11, and our options for C2 are 19 or 23. 23 is the only number that creates a prime sum no matter what C1 or C3 are. C2 is 23, and A2 is 19.

Step 4: The clue for A2 says “The sum of the right most column is greater than the sum of the bottom row”.

This tells us that the number in C1 is the larger of the two pairs, so either 11 or 13, while A3 is either 5 or 3, respectively. Combining all the rules revealed before – only row 3 sums to a prime, only column C sums to a prime, the diagonals both sum to 23 – results in A1 being 5, A3 being 3, C1 being 13, and C3 being 11.