Sujiko

Sujiko is a logic-based, combinatorial number-placement puzzle created by Jai Gomer of Kobayaashi Studios.

A completed Sujiko puzzle.

The puzzle takes place on a 3x3 grid with four circled number clues at the centre of each quadrant which indicate the sum of the four numbers in that quadrant. The numbers 1-9 must be placed in the grid, in accordance with the circled clues, to complete the puzzle.

Sujiko is featured in UK newspapers including The Times, The Telegraph, and The Sun.

Sujiko properties

File:Sujiko cells 2.jpg

Yellow equals middle plus 45

The puzzle has a mathematical property, often ignored by puzzle setters, that can sometimes make supposedly-difficult examples somewhat trivial, as follows:

The sum of two diagonally-opposite CORNER numbers and the two TOTALS on the OTHER diagonal (e.g. the yellow numbers in the second image) is always equal to the MIDDLE number, plus 45.

In letters:
(a + i) + (T2 + T3) [= P, say]
= (c + g) + (T1 + T4) [ = Q, say]
= e + 45

E.g. from the completed Sujiko Puzzle shown,
(4 + 9) + (18 + 15) = 46
(8 + 6) + (17 + 15) = 46
And 1 + 45 = 46

Proof, for P:

T2 = b + c + e + f
T3 = d + e + g + h

So P = (a + i) + (T2 + T3) can be rewritten as
P = (a + i) + (b + c + e + f) + (d + e + g + h)
Or, rearranging,
P = (a + b + c + d + e + f + g + h + i) + e

But (a + ... + i) is just the sum of all the numbers 1 to 9 in some order, i.e. 45

So P = 45 + e

Q is trivially obvious, by symmetry (or can be similarly proved from first principles).