## Number square puzzle

This puzzle was outside a coffee shop at my uni – what number replaces the question mark:

30 16 11 35
20 15 12 14
40 18 14 17
60 14 12 ?

Submitted by Unknown.

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## 13 Comments.

1. Jordan

Could the answer be provided?
Thanks! 7 7

2. Tobbzn

There are always infinitely many patterns in such tasks. I can find an arbitrary polynomial of degree n that passes through all those points, and “deduce” the last one from that. 2 4

3. Glenn

One possible answer is ? = 0. The sum of the columns is 150, 63, 49 and 66 which sum to 328.

The sum of the rows is 92, 61, 89 and 86, which sum to 328.

The value of 0 for the ? would make the sum of the columns and rows equal. However, it would completely mess up the sum of the 2 diagonals which are 125 and 59.

The answer just depends on the arbitrarily decided upon goal.

Hmm, I wonder if there is a value for the ? that would make the determinant of the 4×4 matrix = 0? Not sure if that is possible or not. 1 3

• Laura

If you add all the numbers, it gives you 328 as well. Maybe that’s a help? 3 3

4. Mark

@Glenn:

The sum of the rows and the sum of the columns both equal the sum of all the numbers. So they will always be the same.

The determinant of a matrix of which one element is a variable, is a linear function of that variable, say a*x+b. So it can always be made 0, unless a is 0 and b is not 0. 2 1

5. Glenn

@Mark,
Dang. I screwed up then. I thought it was too easy. As far as the determinant goes, I was being lazy. A 4×4 determinant is a pain in the rear.

Must think harder. 0 1

• mikkel

the determinant gives 380*?+3620 0 1

• David

Which makes ? negative, so that probably will not work. 0 1

6. David

Might the solution have to do with the suspicious fact that all the numbers in the first column are multiples of ten? 0 0

7. Opti

It must be > 12 to preserve row orders [Decrease, decrease, increase]

We have 11-12 and 14 – 18 present, so I would say:
?=13. 0 1

8. Magrathea

Graaaaaphic calculator. Can find this, determinant.

I’ll work on this as I’m drinking beer this afternoon. 0 2

9. Morten

I would say 33.
6-1 (the number just after 6) = 5(last figure in 35)
5-1 (the number just after 5) = 4(last figure in 14)
etc.
i.e. 4-1 = 3 (the last figure in the empty field.)

To get the other “3”, I did the following:

3 (in 35) multiplied by 1 (i.e. the figure just before) is 3 (in 30, the first figure)
1 (in 14) multiplied by 2 (i.e. the figure just before) is 2 (in 20) etc.
I.e. to get 6 (in 60) I multiplied 2 by 3 which was the other missing number in the empty field.

Does that make sense to anyone?

/Morten 2 1

10. Robert

I would say 53.
Why?

30 16 11 35 — 30+16-11=35 — 11 replace 11
20 15 12 14 — 20+15-21=14 — 12 replace 21
40 18 14 17 — 40+18-41=17 — 14 replace 41
60 14 12 53 — 60+14-21=53 — 12 raplace 21

🙂 6 2