# Fill the grid with numbers to make all four equations true

Using all the numbers {1,2,3,4,5,6,7,8}, each exactly once, fill each empty white square with a number so that all the horizontal and vertical equations are true. The grey squares are not used.

The filled grid:

Explanation:

The 1 cannot be part of a multiplication or division without causing duplicate numbers, and cannot be the result of an addition. The only remaining option is at the start of the second row.
Only 2 can work as a divisor, because anything larger will make the division equal 2 and then cause problems with the multiplication in the second column becoming too large.
The 8 now must go in the bottom of the second column, as anywhere else rapidly leads to numbers becoming too large.
The rest then follows easily.

The first row has very few possibilities: 8,2,4; 8,4,2; 6,2,3; 6,3,2. But 8 as the dividend doesn't allow another addend in the first column, so it's 6,3,2 or 6,2,3. The other addend in the first column can't be 2 or 3, then, and 4 is too large, so it's 1. That means that the other multiplicand in the second column is not 1, 2, or 3, so it must be 4 to keep the product small enough: and then the first multiplicand is 2. So we get

6 2 3
1 4 5
7 8

• I like Jaap Scherphuis's solution better than my own, because it combines "what can go in this cell?" and "where can I put this digit?" whereas I just used the former. Commented Sep 3 at 7:06

Consider

the middle number in the first row. It equals the ratio between the other numbers in the first row, and also between the other numbers in the second column. 1 is inadmissible, because the numbers are all distinct, and every number larger than 2 has at most one pair of numbers greater than one with that ratio, so 2 goes in that spot.

3 and 4 now occupy the top right and central spaces, and 6 and 8 occupy the top left and bottom middle spaces, leaving 1, 5, and 7 to be placed. These must go in the center left, center right, and bottom left spaces, respectively, and the rest follows.

 6 2 3
1 4 5
7 8