How to Solve Mathrax Puzzles

Mathrax was first published on the German puzzle website janko.at. The goal is to fill in numbers from 1 to X, where X is the number of cells in one side of the grid, such that no row or column contains any duplicate numbers.

In circles are arithmetic clues with a number and an operator, or the letters E or O.
If the circle contains an E or O, the numbers in all four of the cells surrounding it are all even, or odd, respectively.
If the circle has an operator, the number is the result when that operator is applied to each pair of diagonally adjacent digits.

Mathrax puzzles will not always start you with pre-filled cells. When that happens, just start out with pencil marks of the possible numbers (in this grid, it would be 1-6), and eliminate those around the circle clues that cannot produce the required result with the operator given.

However, in this easy example puzzle, we can make a few swift deductions. Here, we have a pair of cells that must multiply to result in 6. a 3 is given, and we know that 3x2=6, so the empty cell must be a 2.

Next, we have a circle with an addition operator that must result in a sum of 9. As you can see, 5+4 is already given.

Since 3+6=9, we can fill in the empty yellow square with a 6.

Remember, when calculating a result in a circle, it is always the diagonally adjacent cells that are used, and the result must occur when using the pair of numbers in both of the two directions.

This circle requires a subtraction resulting in 4. Because 6 is the highest number allowed in the grid, we cannot use 10-6 or 9-5, as both use a number that is not permitted.

So it must be that in the yellow cells, we’re subtracting from 6 to get 4, which we know is 6-2. In the blue cells, 5-1=4, so we now know to place a 1 in the blue space, and a 2 in the yellow one.

We’ll quickly skim through a few more easy calculations.

Yellow: 1+6=7, and in blue, 6+1=7.

The other technique you will be using to solve a Mathrax puzzle is the same scanning method common to puzzles like Sudoku. Simply look along rows and columns and find numbers which aren’t already used.

In the green row, the only number option left is a 1.

You can also scan in two directions, since a number can’t repeat in either the same row or column.

Here, the only numbers left in the yellow row are 4 and 5, and the blue column only has 3 and 4 remaining.

Since 4 is the only number missing from both row and column, it must be what goes in the green cell. After that, the other empty space in the yellow row will be 5, while the 3 goes at the top of the blue column.

Before entering the last handful of numbers, I wanted to focus briefly on this circle clue that doesn’t have an operator. The letter O means that all four cells around it must be odd numbers. (if it were an E, they would all be even).

So we can see that the last two number options for this column are 1 and 4. Because of the O clue, we know that the 1 has to go in the highlighted cell, and the 4 will go at the top of the column.