Magic/Latin square is a square filled with numbers/letters, which are all distinct in each row and column. Sudoku is 9*9 magic square with additional constraints (for each 3*3 subsquare).

"Knut Vik design" is a square, where all (broken) diagonals has distinct numbers.

This is diagonal of 5*5 square:

. . . . * . . . * . . . * . . . * . . . * . . . .

These are broken diagonals:

. . * . . . . . * . . . . . * * . . . . . * . . .

* . . . . . . . . * . . . * . . . * . . . * . . .

I could only find 5*5 and 7*7 squares using Z3, couldn't find 11*11 square, however, it's possible to prove there are no 6*6 and 4*4 squares (such squares doesn't exist if size is divisible by 2 or 3).

from z3 import * #SIZE=4 # unsat #SIZE=5 # OK #SIZE=6 # unsat SIZE=7 # OK a=[[Int('%d_%d' % (r,c)) for c in range(SIZE)] for r in range(SIZE)] s=Solver() # all numbers must be in 1..SIZE limits for r in range(SIZE): for c in range(SIZE): s.add(And(a[r][c]>=1, a[r][c]<=SIZE)) # all numbers in all rows must be distinct: for r in range(SIZE): # expression like s.add(Distinct(a[r][0], a[r][1], ..., a[r][last])) is formed here: s.add(Distinct(*[a[r][c] for c in range(SIZE)])) # ... in all columns as well: for c in range(SIZE): s.add(Distinct(*[a[r][c] for r in range(SIZE)])) # all (broken) diagonals must also be distinct: for r in range(SIZE): s.add(Distinct(*[a[(r+r2) % SIZE][r2 % SIZE] for r2 in range(SIZE)])) # this line of code is the same as previous, but the column is "flipped" horizontally (SIZE-1-column): s.add(Distinct(*[a[(r+r2) % SIZE][SIZE-1-(r2 % SIZE)] for r2 in range(SIZE)])) print s.check() m=s.model() for r in range(SIZE): for c in range(SIZE): print m[a[r][c]].as_long(), print ""

5*5 Knut Vik square:

3 4 5 1 2 5 1 2 3 4 2 3 4 5 1 4 5 1 2 3 1 2 3 4 5

7*7:

4 7 6 5 1 2 3 6 5 1 2 3 4 7 1 2 3 4 7 6 5 3 4 7 6 5 1 2 7 6 5 1 2 3 4 5 1 2 3 4 7 6 2 3 4 7 6 5 1

This is NP-problem: you can check this result visually, but it takes several seconds for computer to find it.

We can also use different encoding: each number can be represented by one bit. 0b0001 for 1, 0b0010 for 2, 0b1000 for 4, etc. Then a "Distinct" operator can be replaced by OR operation and comparison against mask with all bits present.

import math, operator from z3 import * #SIZE=4 # unsat SIZE=5 # OK #SIZE=6 # unsat #SIZE=7 # OK #SIZE=11 # no answer a=[[BitVec('%d_%d' % (r,c), SIZE) for c in range(SIZE)] for r in range(SIZE)] # 0b11111 for SIZE=5: mask=2**SIZE-1 s=Solver() # all numbers must have form 2^n, like 1, 2, 4, 8, 16, etc. # we add contraint like Or(a[r][c]==2, a[r][c]==4, ..., a[r][c]==32, ...) for r in range(SIZE): for c in range(SIZE): s.add(Or(*[a[r][c]==(2**i) for i in range(SIZE)])) # all numbers in all rows must be distinct: for r in range(SIZE): # expression like s,add(a[r][0] | a[r][1] | ... | a[r][last] == mask) is formed here: s.add(reduce(operator.ior, [a[r][c] for c in range(SIZE)])==mask) # ... in all columns as well: for c in range(SIZE): # expression like s,add(a[0][c] | a[1][c] | ... | a[last][c] == mask) is formed here: s.add(reduce(operator.ior, [a[r][c] for r in range(SIZE)])==mask) # for all (broken) diagonals: for r in range(SIZE): s.add(reduce(operator.ior, [a[(r+r2) % SIZE][r2 % SIZE] for r2 in range(SIZE)])==mask) # this line of code is the same as previous, but the column is "flipped" horizontally (SIZE-1-column): s.add(reduce(operator.ior, [a[(r+r2) % SIZE][SIZE-1-(r2 % SIZE)] for r2 in range(SIZE)])==mask) print s.check() m=s.model() for r in range(SIZE): for c in range(SIZE): print int(math.log(m[a[r][c]].as_long(),2)), print ""

That works twice as faster (however, numbers are in 0..SIZE-1 range instead of 1..SIZE, but you've got the idea).

Besides recreational mathematics, squares like these are very important in design of experiments.

The files: https://github.com/dennis714/yurichev.com/tree/master/blog/knut_vik

Further reading:

- A. Hedayat and W, T. Federer - On the Nonexistence of Knut Vik Designs for all Even Orders (1973)
- A. Hedayat - A Complete Solution to the Existence and Nonexistence of Knut Vik Designs and Orthogonal Knut Vik Designs (1975)

→ [list of blog posts, my twitter/facebook]

Please drop me email about any bug(s) and suggestion(s):