Sum Sudoku

Logic Level 3

Here are the rules of sum Sudoku :

Each and every column, row, and $2\times2$ box contains distinct numbers from 1 to 4.
The white circle marked at four adjacent cells indicate that the sum of upper-left and bottom-right cells is the sum of upper-right and bottom-left cells.

What can we conclude about the circle sums in the puzzle?

Example:
The setup in the top-left box is valid since $2 + 3 = 1 + 4 = 5$ . However, the four cell values at the center are invalid since $4 + 2 \neq 3 + 1$ .

All sums must be the same All but the center circle must be the same The center circle sum has multiple values that work

1 solution

Michael Huang
Aug 18, 2017

Let's look at the cells adjacent to the center circle.

$Since $a \neq b$, it is impossible for the center sum to exist.$ Since $a \neq b$ , it is impossible for the center sum to exist. Suppose that it is possible to label common digit in adjacent cells at the center. Along one of the diagonals, label $a$ and $b$ , where $a + b = 5$ (since the circle sum within the box is unique). Value-searching, we have the setup as shown above. However, since the pairs of $a$ 's and $b$ 's are adjacent to the center circle, this shows that $2a = 2b$ , which is impossible since this violates the first rule.

Therefore, no opposite cells can have common values. Because $5$ is the only number that can be expressed as two different integer sums, $\boxed{\text{all circle sums must be the same}}$ .

It is easy to notice that since the puzzle is symmetric about the center circle, we can either reflect or rotate it to determine other solutions!

Sum Sudoku

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