Changing the measurement ledger changes the accepted grid without changing the reconstruction rule. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
First measurement set accepts the base grid
Rows 3, 7; columns 4, 6; diagonal 5 accept grid 1, 2; 3, 4.
1 + 2 = 3 3 + 4 = 7 1 + 4 = 5 1+2=3\quad3+4=7\quad1+4=5 1 + 2 = 3 3 + 4 = 7 1 + 4 = 5 Tomography solve A tiny reconstruction is accepted only when the measurements pin it. 1 2 3 4 0 0 0 0 0 1
Second set accepts the swapped grid
Rows stay 3, 7 while columns are 6, 4. The accepted grid is 2, 1; 4, 3.
2 + 4 = 6 1 + 3 = 4 2 + 3 = 5 2+4=6\quad1+3=4\quad2+3=5 2 + 4 = 6 1 + 3 = 4 2 + 3 = 5 Tomography solve A tiny reconstruction is accepted only when the measurements pin it. 2 1 4 3 0 0 0 0 0 1
Third set accepts the larger grid
Rows 5, 9; columns 6, 8; diagonal 7 accept the larger grid with zero residuals.
2 + 3 = 5 4 + 5 = 9 2 + 5 = 7 2+3=5\quad4+5=9\quad2+5=7 2 + 3 = 5 4 + 5 = 9 2 + 5 = 7 Tomography solve A tiny reconstruction is accepted only when the measurements pin it. 2 3 4 5 0 0 0 0 0 1