Suppose you want to fill the 5x5 square below with the following restrictions:
(1) The numbers 1-5 must appear in every row and column;
(2) The sum of the numbers in gray boxes is 2 (mod 6).
What number must go in the center square?
Don't care about the white squares, just care about the grey squares.
We can fill the horizontal grey row with 1 2 3 4 5
The remaining four grey squares will contain 4 numbers from the list 1 2 3 4 5
Since is 2 (mod 6)
we just guess and check
1 + 2 + 3 + 4 + 5 = 15
That is the sum of the center horizontal row.
We then can guess and check the sum of the vertical row plus 15 to see if it fits 2 mod ( 6)
Lets say the center square is 1?
Then 29 / 6 remainder is 5
center square is 2?
Then 28 / 6 remainder is 4
Center square is 3?
then 27/6 remainder is 3
Center square is 4?
then 26/6 remainder is 2
Center squar eis 5?
25/6 remainder is 1
We see by guess and checking that the answer is 4, four should go into the center square
I actually used guess and check in this problem XD XD, I only understood the format meaning of modulus I actually barely even know how to apply it.
Suppose you want to fill the 5x5 square below with the following restrictions:
(1) The numbers 1-5 must appear in every row and column;
(2) The sum of the numbers in gray boxes is 2 (mod 6).
What number must go in the center square?
Let the number go in the center square \(= \mathbf{x}\)
Let the sum of the numbers \(1+2+3+4+5 = 15\)
Let the sum of the numbers in gray boxes \(= 15 + (15-x)\)
Then:
\(\begin{array}{|rcll|} \hline 15 + (15-x) & \equiv & 2 \pmod 6 \\ 30-x & \equiv & 2 \pmod 6 \quad | \quad 30 \pmod 6 = 0 \\ 0-x & \equiv & 2 \pmod 6 \\ -x & \equiv & 2 \pmod 6 \quad | \quad \cdot (-1) \\ x & \equiv & -2 \pmod 6 \\ x & \equiv & -2+6 \pmod 6 \\ x & \equiv & 6-2 \pmod 6 \\ \mathbf{x} & \equiv & \mathbf{4 \pmod 6} \\ \hline \end{array}\)
The number 4 must go in the center square.