Fun with grids

Here is a calendar page for August 2025. A friend secretly picks any 2 × 2 square of four dates, adds them, and tells you only the total. You name all four dates.

How? A 2 × 2 square on a calendar is never four free numbers. Once you know the top-left date, the other three are fixed.

Let the top-left date be a. The one next to it is a + 1. The one below it is a + 7, because a week has seven days. The fourth is a + 8. (On a real calendar: 15, 16, 22, 23.)

So the sum is a + (a + 1) + (a + 7) + (a + 8) = 4a + 16: one unknown, not four.

Your friend says 36. Then 4a + 16 = 36, so 4a = 20, so a = 5. The square is 5, 6, 12 and 13. Announce them and enjoy the surprise.

You can invent your own version. A 3 × 3 square, an L shape, two dates on a diagonal: each has its own expression in a. On a grid whose rows go up by 9 instead of 7, the date below becomes a + 9 and the sum becomes 4a + 20. Same method, different fixed number.

Algebra grids use the same idea, with shapes standing for numbers. In each row, the last column is the sum of the shapes to its left, so each row is an equation.

Find a row that tells you one shape on its own (for example, three of the same shape with a total: three circles making 27 means one circle is 9). Put that value into the next row to find the second shape, and carry on. It is the pyramid method again: name what you do not know, and let the rules give you the equations.