# Find the value of '?' in 4x4 table of numbers [closed]

Solve for ?

50 15 10 60
45 31 5 87
40 22 42 ?
9 4 3 14
• Hi William welcome to PSE could you please give us the source of this puzzle?
– PDT
Commented Jun 26 at 14:55

The entry is, or rather may be $$85$$, with the following way to guess it. Take the first and the third columns. Add them. To the result add four times the second column. We obtain: $$\begin{bmatrix} 50 + 4\cdot 15+10\\ 45 + 4\cdot 31+5 \\ 40 + 4\cdot 22+42\\ 9 + 4\cdot 4+3 \end{bmatrix} = \begin{bmatrix} 120\\ 174 \\ 170\\ 28 \end{bmatrix} = 2\cdot \begin{bmatrix} 60\\ 87 \\ \bbox[yellow]{\ 85\ }\\ 14 \end{bmatrix} \ .$$ There is a match with the last column at three places...
Note: This is the value that makes the given $$4\times 4$$ matrix have linearly dependent rows and columns. Well, the combination of columns is still "nice", so this is a puzzle plus for the value. (But one should never mention, here or elsewhere, the combination of the rows, it needs the weights / coefficients $$1609$$, $$805$$, $$655$$, $$-15875$$, or common multiples, except for the case when still not afraid to endure the consequences.)