# Cartouche puzzles

The most advanced concept used in these puzzles is distributing and combining like terms with variables, but these puzzles can be designed to focus on less advanced concepts too. For example, you could design a puzzle where the most advanced concept is dividing one two-digit number by another.

Note: A puzzle involving multiplication is likely easier than a puzzle involving just addition and subtraction. Consider that 10 can be written in many ways as the sum of 2 single-digit numbers: \(1 + 9,\) \(2 + 8,\) \(\ldots,\) \(9 + 1,\) but only two ways as the product of two single-digit numbers: \(2 \cdot 5,\) or \(5 \cdot 2.\)

Strategy 1:

Look for multiplication regions (red) that are prime, such that the prime is greater than any remaining possibility. For example, in a grid where only 1-4 are allowed, a red region with a prime greater than 4, tells us that one of the red digits is 1. Here's an example:

Strategy 2:

Just as in Sudoku, writing in the remaining possibilities can be useful. For example, referring once more to the image above, the addition region (blue) must add to 7, using 1-4, and thus, the only possibilities are 3 + 4 = 7, and 4 + 3 = 7. These can be noted by writing 34 and 43 somewhere in the blue region.

Strategy 3:

If a number is not prime, then consider its factors. For example, in the figure below, concentrate on the solely red square.

8 has factors 1, 2, 4, and 8. But if we're only allowed 1-4, then 8 is not a possibility. Suppose we choose 1. Then the blue digits have to add up to 8, which means both would have to be 4, which is impossible. Suppose we put 4 in the solely red square, instead of 1. Then the blue square have to add up to 2, which is also impossible. Thus, the solely red square must be 2, as that's the only factor of 8 that remains. This tells us the blue squares must add to 4, and so they must be 1 and 3, in some order.