I'm solving a missing number in square problem. I have tried multiple ways but haven't been able to find any common rule. My attempts are-

  1. $((5+7+6)+1) \times 6=114$
  2. $4 \times ((3+5)+1) = 36$

Please refer to Question number $39$ in below image. image link


The following rule gives a unique answer which is among the choices. Whether it is the rule they were looking for....who knows.

The rule is (not including the bottom row):

Multiply all numbers in a column. Add to this value the sum of all numbers in the column which are unique. Divide the resulting value by $2$ to get the value at the bottom row.

This gives the answer $4$, i.e. A.

Let's check if this rule works for each column.

Column 1, value in last row:

$$Value = \frac{3*5*4+3+5+4}{2} = \frac{72}{2} = 36$$

Column 2, value in last row:

$$Value = \frac{4*4*4}{2} = \frac{64}{2} = 32$$

Column 3, value in last row:

$$Value = \frac{5*7*6+5+7+6}{2} = \frac{228}{2} = 114$$


  • $\begingroup$ Is it really reasonable to expect students to figure this out? And what the heck is this supposed to teach anyway? ... Okay... sorry... sorry... it's not your fault. You didn't write the question. +1 for figuring out what stumped me. $\endgroup$ – fleablood Feb 2 '17 at 21:06
  • 1
    $\begingroup$ @fleablood No, it is not at all reasonable. It took me several hours of pondering and I love puzzles! Still, we don't know the context. Is this perhaps an intelligence test which in the last few questions tries to differentiate between the genious and the merely very bright? $\endgroup$ – Jens Feb 2 '17 at 22:11
  • $\begingroup$ @Jens brilliant solution! thanks $\endgroup$ – iajnr Feb 2 '17 at 22:59

I can not ####ing believe they still teach math this way!

The hallmark of an intelligent student should be the realization that assumed rules use assumptions that may or may not be valid and therefore any conclusion can not be certain. Therefore the development of any good mathematician is to stop making assumptions. These "find the rules" do the exact opposite.

And any mathematician eventually learns that rules may be constructed for any finite set of points... which means I can put in ANY answer into the square an define a rule that will fit them all. (That was actually the tongue in cheek intent of Jan's answer--- you can algebraicly create a polynomial that will give any answer you want.) Indeed this is fundamental to the concept of "function". The "mathematicians" who make these exercises should know that but instead they teach absolute falsehoods and sloppy thinking to innocent students. It ####es me off so much!

Okay.... calm down.

It's just a puzzle. And puzzles are fun. And the goal is to find a simple rule (Jan's rules are not "simple".) ....

But that's not MATH! That's puzzles. "Simple" is not well-defined and it isn't at all clear what the consider simple. And to find a rule counts on intuition and .... when you get down to it, you can not teach intuition. And if you try, you end up teaching students rote rules that leave them totally helpless and with recourse if the question takes even the slightest altercation.

There was study done in the 80's about how students do word problems. The gave students the question: "Malcolm left has a car. He drove is car 7 miles. Then he turned right and drove 2 miles. How many miles did Mr. Left drive?" Many students answered ... 5. Why? Because the man's name was Mr. Left and "left" means subtraction, of course!

This type of math teaching is no different.

Anyway... So I look at your square.... I don't see it either. I think the line with 36, 32, and 114 and so much larger than the other rows that the rule must lie in the columns. Somehow 3,5,and 4 make 36; ?,4,4 make 32, and 5, 7 and 6 make 114. But I'm damned if I see it. (I do see that 4 divides 36. 4 divides 32 and 6 divides 114 but I don't see any way that relates.)

So don't worry. If you can't do this it doesn't mean you can't do math. It means you can't do puzzles. ... Well, no it doesn't. It means you can't do this puzzle. And neither can many other people here.

  • $\begingroup$ im a student of grade 8th, yes i agree with you that this is hardly mathematical but yes this is a small portion of our syllabus. @fleablood $\endgroup$ – iajnr Feb 2 '17 at 23:00

There is nothing mathematical in your question except that numbers are being added. There probably exists infinity of solutions to your problem. One way to solve it is to write the 'rule' your question mentions (note that the rule is completely arbitrary) as $$\begin{aligned} 3a+5b+4c=36\\ xa+4b+4c=32\\ 5a+7b+6c=114\\ \end{aligned}$$ and solve this system of equations for fixed $x$. The solution is $a=\frac{124}{4-x}$, $b=-120+\frac{124}{4-x}$ and $c=159-\frac{248}{4-x}$. When, for example, $x=1$ this collapses to $a=124$, $b=4$ and $c=-89$.

  • $\begingroup$ Your "rule" appears to allow all answers except A. In what sense could this then be the rule used? $\endgroup$ – Jens Feb 2 '17 at 16:46
  • $\begingroup$ @Jens I completely missed the multiple choice options. My rule allows for all with $x=-27$, $x=-50/3$, $x=-88/9$ and $x=-23/2$. Only positive integers allowed for $x$? The question did not say. I thought my scepticism about the question was apparent from the first sentence of my answer. $\endgroup$ – Jan Feb 2 '17 at 16:51
  • $\begingroup$ @Jan I agree that there is of course nothing deductive about this ... and that in principle there are infinitely 'solutions' ... we're probably just looking for a 'nice' rule. Moreover, it's not even clear that this rule involves addition ... or that the bottom value is somehow the result of some function over the 3 values above it in the same column ... any of the numbers anywhere in the grid could be any kind of function of any of the numbers in the grid ... you could probably even work out some kind of self-reference! $\endgroup$ – Bram28 Feb 2 '17 at 17:00

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