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Incubator Q&A Fill the 4x4 grid with numbers to make eight arithmetic progressions

In some Mathematical puzzles, there are several unknown values. In this particular puzzle, there are 12 unknown numbers that are to be placed in the 12 empty squares. We could attempt to solve th...

posted 1mo ago by will.octagon.gibson‭

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#1: Initial revision by

will.octagon.gibson‭ · 2026-08-07T16:53:23Z (about 1 month ago)

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In some Mathematical puzzles, there are several unknown values.  In this particular puzzle, there are 12 unknown numbers that are to be placed in the 12 empty squares.

We could attempt to solve this puzzle by using 12 variables, one for each unknown number, plus a number of equations relating these 12 variables.  But this would be a tedious way to determine the numbers for the empty squares.

With a bit of ingenuity, we can solve this puzzle using only 1 variable.

<details>

<summary>Solution</summary>

### Step 1:

Let the common difference between consecutive numbers in the top row be **d.**

So the grid is:

```text
27     27+d   27+2d  27+3d
                     55
              45
       25
```

### Step 2:

In the rightmost column the common difference between consecutive numbers is:

55-(27+3d) = 28-3d.

So the number below the 55 is:

55+(28-3d) = 83-3d.

So now the grid is:

```text
27     27+d   27+2d  27+3d
                     55
              45     83-3d
       25
```

### Step 3:

In the third row the common difference between consecutive numbers is:

(83-3d)-45 = 38-3d.

So the number to the left of the 45 is:

45-(38-3d) = 7+3d.

So now the grid is:

```text
27     27+d   27+2d  27+3d
                     55
        7+3d  45     83-3d
       25
```

### Step 4:

In the second column, the first and third numbers are:

27+d and 7+3d.

**Twice** the common difference between consecutive numbers in the second column is:

(7+3d) - (27+d) = -20+2d.

So the common difference between consecutive numbers in the second column is:

(-20+2d)/2 = -10+d.

So the second number in the second column is:

(27+d) + (-10+d) = 17+2d.

So now the grid is:

```text
27     27+d   27+2d  27+3d
       17+2d         55
        7+3d  45     83-3d
       25
```

### Step 5:

Now all our hard work is about to yield a great result.

Using the common difference between consecutive numbers in the second column, the number at the bottom of the second column (25) is also equal to:

(7+3d) + (-10+d) = -3+4d.

Therefore, 25 = -3+4d.

Solving we get d = 7.

Using this value of d, we can calculate some of the grid numbers.

So now the grid is:

```text
27     34     41     48
       31            55
       28     45     62
       25
```

### Step 6:

Finally, we can now fill in the rightmost column and then the rows to get:

```text
27     34     41     48
19     31     43     55
11     28     45     62
 3     25     47     69
```

</details>

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