Logic Masters Deutschland e.V.

(This is the fourth puzzle in the inverse sandwich series. The previous one needed bifurcation when it was initially posted, so if you tried to solve that one and couldn't, it's fixed now!)

1 and 9 cannot be a chess knight's move from an identical digit. The grid is toroidal, which means for the purpose of the antiknight constraint and clues, rows 1/9 and columns 1/9 are adjacent. Clues outside the grid give the sums of the digits that are orthogonally adjacent to either the 1 or 9 in the row or column. The solver must determine which clues are for ones, and which are for nines. Where two clues appear, they give the sums for both the 1 and 9, and the order in which the sums appear (left to right in a row, top to bottom in a column, sums that wrap around the edge are ordered according to where the 1/9 appear). Some clues are replaced with question marks, which represent unknown numbers. The equal sign gives the relationship between two clues. Digits in cages sum to the number in the upper left corner.

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