9
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Consider the following 4x4 grid :

972    9    5   55
 18   22    x   28
 50   24   25   26
  7  400   52    4

Find $x$. However, as the title suggests, there are multiple solutions. You have to find them all and explain why.

Bonus question: Multiple grids can give the same set of solutions. Find how many different grids have the same solutions as the grid above.

Hint #1:

The number of solutions is somewhere between 6 and 17.

Hint #2:

This might be related to magic squares

Hint #3 (this one helps a lot, but you can still solve the puzzle without seeing this hint. If you want a real challenge, don't look at this one.):

My 4x4 grid was entirely created using the magic square in hint#2.

More hints will be given over time.
Good luck.

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  • $\begingroup$ I tried many solution but,none of them works .Can you provide some hints :) $\endgroup$ – Swati Mar 29 at 13:05
  • $\begingroup$ Hint#2 added. Should be easier to solve now. $\endgroup$ – classicalMpk Mar 29 at 14:02
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There are

10

possibilities.

Explanation:

If we factor the numbers in the grid, we get (take the first row as example):

972 = 2*2*3*3*3*3*3; 2+2+3+3+3+3+3 = 19 = 16+3
9 = 3*3; 3+3 = 6 = 3+3
5 = 5; 5 = 2+3
55 = 5*11; 5+11 = 16 = 13+3

That is, sum of (number of the grid factored) = (corresponding number of the magic square) + 3

Therefore, the x in the grid corresponds to 11 in the magic square
--> sum of (x factored) = 14, and 14 has 10 prime partitions

The 10 possibilities are:
14
= 2+2+2+2+2+2+2 --> 2*2*2*2*2*2*2 = 128
= 2+2+2+2+3+3 --> 2*2*2*2*3*3 = 144
= 2+3+3+3+3 --> 2*3*3*3*3 = 162
= 2+2+2+3+5 --> 2*2*2*3*5 = 120
= 3+3+3+5 --> 3*3*3*5 = 135
= 2+2+5+5 --> 2*2*5*5 = 100
= 2+2+3+7 --> 2*2*3*7 = 84
= 2+5+7 --> 2*5*7 = 70
= 7+7 --> 7*7 = 49
= 3+11 --> 3*11 = 33

For the bonus question:

According to the rules of constructing the grid,
The possible numbers of grids
= product of (possible numbers of each square)
= product of (# of prime partitions of (number in the magic square)+3)
and 4, 5, ..., 19 has 1, 1, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23 different prime partitions each
Therefore the possible number of grids = 1*1*2*3*3*4*5*6*7*9*10*12*14*17*19*23 = 1698376377600
or 169837637760 excluding the "x" square

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  • $\begingroup$ Exactly ! You just forgot the second partition of 5 (2;3, 5), so for the bonus question you will have to multiply your answer by 2. Did you find these partitions by hand or did you use a program? $\endgroup$ – classicalMpk Apr 9 at 10:09
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    $\begingroup$ I used this website for partitioning the numbers. $\endgroup$ – naldjuno Apr 9 at 10:58
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Looking at hint i think we need to either add both grid number or subtract or assign the number given in particular order .I am not sure if i am right or not but ,here is the solution :

Solution 1 : If we add these both grid we get :

972    9    5   55           |    16   3  2  13
 18   22    x   28           |     5  10  11  8
 50   24   25   26           |     9   6   7 12
 7  400   52    4            |     4  15  14  1
This grid :
988 12 7 68 
 23 32  x 36 
 59 30  32 38 
 11 415  66 5
Here ,we get x value = 27
because ,  if we add column 3 values i.e : 7 + 32 + 27 = 66
if we add diagonal we get i.e: 11 + 30 + 27 = 68
And if do same to 2nd row we get : 32 + 27 - 23 = 36

Solution 2 :

Now if we assign values according to second grid 1,2 ..16 then the grid will look like below :

972 7 5 52
18 28 x 25
26 22 24 50
9 400 55 4
Now if take second row  : 28 - 18 = 10 + 16 =26 - 1= 25 
Taking third colmnn : 16 * 5 - 24= 56 -1 = 55
So observing here value of x will be 16

Solution 3:

If we take original grid :

972  9    5   55
18 22 x 28
50 24 25 26
7 400 52 4
If we take second row to get value 28 we take x = 24 i.e: 22 + 24 = 46 - 18= 28. Now same we can do with diagonal i.e : 7 + 24 + 24 = 55. So , x can be 24 also.

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  • $\begingroup$ Adding the two grids will not help you finding the solution. The magic square was added as a hint : therefore it is not needed to solve the puzzle. It might still be hard ; I will add hint#3 in a few minutes. $\endgroup$ – classicalMpk Apr 5 at 13:54
  • $\begingroup$ with multiple grids you mean a magic square can be arrange in multiple ways and result will be same ? $\endgroup$ – Swati Apr 8 at 4:16
  • $\begingroup$ If you're talking about the bonus question, then 'multiple grids' means you can change some numbers and the solutions will still be the same. Everything should be clearer when you solve the main question. If you want a tip, then focus on the small numbers (1,2,3,4) of the magic square and see what they became in my 4x4 grid. $\endgroup$ – classicalMpk Apr 8 at 9:52
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Wrong: Partial answer (3 values for x):

Solution 1: x could be 12,
Because the diagonal $9, x, 26$ is exactly half of the diagonal $18, 24, 52$, thus $x$ could $12$.

Solution 2: x could be 23,
Because the four values in the square middle:
$22$ $x$
$24$ $25$
form a sequence: $22, x, 24, 25$, thus x could be $23$.

Solution 3: x could be 2
Because the third column consist of combinations of numbers with $5$ and $2$ but misses the number $2$.

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  • $\begingroup$ Nice try ! Unfortunately none of these are solutions (the relationships you found were unintentional). Try comparing the image in hint#2 and the grid, it will help you find what the numbers are, and why there are multiple solutions. $\endgroup$ – classicalMpk Apr 3 at 10:52

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