# day 57: sharpening the tools

BTIW:

1. I took time to read the Bible readings for today, yesterday, and Monday.

2. I didn’t use the heaviest weight ever for my push presses, and I strung together 20 double unders.

TINTWO:

1. Communication

from AIME 1988 Problem 06

It is possible to place positive integers into the vacant twenty-one squares of the 5 times 5 square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*).

Solution (general)

First, let the number to be placed in the first column, fourth row. Let the number to be placed in the second column, fifth row. We can determine the entire first column and fifth row in terms of and :

Next, let the number to be placed in the second column, fourth row. We can determine the entire second column and fourth row in terms of , , and :

We have now determined at least two values in each row and column. We can finish the table without introducing any more variables:

We now have a system of equations.

Solving, we find that

.

The number in the square marked by the asterisk is