Letters in a calendar square
Circle four dates in a 2 by 2 square on a calendar. A diagonal joins opposite corners. Add the two dates on each diagonal.
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- Call the top-left date a.
- The date on its right is a + 1.
- A row has 7 dates. The date below is a + 7.
- The diagonal date is a + 8.
- Each diagonal adds to 2a + 8.
Both diagonals give the same total.
The rule
Call the top-left date a. Right of it is a + 1. Below it is a + 7. Across the diagonal is a + 8. Each diagonal adds to 2a + 8.
Try a = 16. The square holds 16, 17, 23 and 24. The diagonals give 16 + 24 = 40 and 17 + 23 = 40. And 2 × 16 + 8 = 40.
A plus shape of five dates works the same way. It has one date in the middle and four around it. Call the middle date c. Above it is c − 7. Left of it is c − 1. Right of it is c + 1. Below it is c + 7. Add all five: c + (c − 7) + (c − 1) + (c + 1) + (c + 7). The −7 and +7 add to 0. The −1 and +1 add to 0. So the five dates add to 5c.
Watch out!
"I tried five squares, so it is always true." Five squares do not prove it. The letter a stands for every square at once. That is why 2a + 8 proves it.