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?

mathleteig Oct 20, 2019

#1**+2 **

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

CalculatorUser Oct 20, 2019

#2

#3**+2 **

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.

CalculatorUser
Oct 20, 2019

#4**+3 **

**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.

heureka Oct 21, 2019