Math Square Help Please

[Oneluckystar10]

Can only use the #'s 1 - 16 once. Thanks

 

[AvgStudent]

I'm assuming PEMDAS doesn't apply, it's strictly left to right, top to bottom.

 

[JeffM]

Tarnation, sixteen Diophantine equations subject to sixteen constraints.

No, no, life is too short!

 

[Oneluckystar10]

I'm assuming PEMDAS doesn't apply, it's strictly left to right, top to bottom.

I think it does but that could also be why I am struggling with it. I know I cannot use the numbers 2, 8, 9, and 11 because they are already in the puzzle. I've been using Guess and Check to get the solution but keep getting stuck.

 

[BigBeachBanana]

Tarnation, sixteen Diophantine equation subject to sixteen constraints.

No, no, life is too short!

Indeed, I don't see an analytical way to solve this besides computer algorithms. The most you can do is odd and even.

 

[BigBeachBanana]

I think it does but that could also be why I am struggling with it. I know I cannot use the numbers 2, 8, 9, and 11 because they are already in the puzzle. I've been using Guess and Check to get the solution but keep getting stuck.

It's because it's not meant for humans to solve IMO.

 

[lookagain]

Tarnation, sixteen Diophantine equations subject to sixteen constraints.

No, no, life is too short!

There are 12 remaining unknowns and a total of
8 equations.

 

[Cubist]

The following row implies that g can only have two possible values...
g*11 - h + i = 27

 

[lev888]

The following row implies that g can only have two possible values...
g*11 - h + i = 27

And one of them does not work.

 

[Cubist]

Finding "g" doesn't seem to help with obtaining the first column's numbers. I think that column, ignoring the order of the subtractions, must be one of:-

1-3-10-16 = -28
1-3-12-14 = -28
4-3-13-16 = -28
4-3-14-15 = -28
5-3-14-16 = -28
6-3-15-16 = -28

Can anyone can see a way to cut these possibilities down (or perhaps see any that I've missed)?

 

[lev888]

Finding "g" doesn't seem to help with obtaining the first column's numbers. I think that column, ignoring the order of the subtractions, must be one of:-

1-3-10-16 = -28
1-3-12-14 = -28
4-3-13-16 = -28
4-3-14-15 = -28
5-3-14-16 = -28
6-3-15-16 = -28

Can anyone can see a way to cut these possibilities down (or perhaps see any that I've missed)?

I think the idea in these problems is to iterate over rows and columns and gradually limit the ranges of all variables. The fact that we could quickly find g is an exception. My approach is to make a table with rows for 1-16 and columns for a-l and fill the cells with Xs whenever a variable-value pairs are eliminated.

 

[Cubist]

I wrote some code to solve this. I can say with certainty that PEDMAS does apply. Also, there's only one possible solution.

Spoiler: numbers in the first column

The first column contains 1, 3, 10, 16 in some order

 

[Oneluckystar10]

I got it solved. Thanks everyone for trying. I just kept doing the guess and check method.

 

[JasonTurner]

Hello,
Looking at the question, I can definitely say that PEMDAS is not possible. To use PEMDAS correctly, always start with parentheses or groups, then exponents, multiplication and division, and finally addition and subtraction. PEMDAS simply refers to many scenarios in which everything follows a set of stages in a predetermined order. P (parentheses) stands for any type of grouping, such as brackets, dashes, and braces, as well as groups suggested by radical and fractional expressions. This is the initial step, and you must solve all groups from the inside out.
Exponents and radicals are both represented by the letter E (exponents). From left to right, execute M (multiplication) and D (division). When there are variables and other notations for products and quotients, however, you should be cautious. Additions and subtractions must be done last, from left to right, as shown by the letters An and S.
This could be done through different approaches.
Answers for -28 as describes before by someone
1 -3 -10 -16 = -28
1 -3 -12 -14 = -28
4 -3 -13 -16 = -28
4 -3 -14 -15 = -28
5 -3 -14 -16 = -28
6 -3 -15 -16 = -28
Let’s see some of the possibilities of -48 from this
1 -9 * 5 -4 = -48
4 -9 * 4 -16 = -48
5 -9 * 4 -17 = -48
6 -9 * 4 -18 = -48
Some of the possibilities of 19 from this
3 +8 -1 +9 = 19
3 +8 -2 +10 = 19
Or in the negative
3 +8 -20 +28 = 19


And you can go around…

 

[Cubist]

I can definitely say that PEMDAS is not possible
...
And you can go around…

I'm not 100% sure what you're saying. Perhaps you're saying that PEMDAS on its own can't be used to find the final solution? If so, then I agree. However, PEMDAS is part of the method required to finding a solution because it's used for the calculation of every row and column result. I used trial and error (backtracking) in conjunction with PEMDAS to find the answer.

Or in the negative
3 +8 -20 +28 = 19

The numbers 20 and 28 aren't allowed in the grid, you can only use the numbers 1-16, and each one is used once.

In case it helps, I include the final solution within the following spoiler...

Spoiler: final solution

 1 -  9 *  5 -  4 = -48
 -    *    -    +
10 +  8 - 14 + 15 = 19
 -    -    +    +
 3 * 11 - 13 +  7 = 27
 -    +    +    +
16 *  2 + 12 +  6 = 50
 =    =    =    =
-28   63   16   32