There's a reason why the puzzle-writer used those 5 digits in the final question, so that you can spot the pattern and find a shortcut. Spotting the pattern is

There's a reason why the puzzle-writer used those 5 digits in the final question, so that you can spot the pattern and find a shortcut. Spotting the pattern is not brute force. To check your formula you have to do some calculations, but that is something you cannot avoid. By going over the numbers, what I mean is confirming that: 1x3 + 2x1 + 3x4 + 4x1 + 5x5 = 1 5x3 + 4x1 + 3x4 + 2x1 + 1x5 = 2 ... You can do that manually or mentally, so it is not what I call brute force. And if you do not spot the 3 1 4 1 5 pattern, or just do not want to use the shortcut provided, there is another way to do it. First let's find the second digit. We only need our known digits and the two lines 2 2 2 2 2 = 1 1 1 2 2 2 = 6 Since we know that the multiplier of the first digit is 3, we can calculate the value of 1 2 2 2 2 = 2 2 2 2 2 - 1 0 0 0 0 = 1 - 3 = 7. Then it follows that the multiplier of the second digit is 1 2 2 2 2 - 1 1 2 2 2 = 7 - 6 = 1. The third digit can be found by changing the line 1 2 3 4 5 = 1 to 3 3 3 4 4 = 3. You can do this since you have the multipliers for the first, second and fifth digits. We already have the line 3 3 4 4 4 = 7, so the third digit is 7 - 3 = 4. For the fourth multiplier, change the line 1 1 1 1 1 = 5 to 2 2 2 1 2 = 0. Compare to 2 2 2 2 2 = 1 and you see that the fourth multiplier is 1. So now you have all the five multipliers, 3, 1, 4, 1 and 5. But we are not necessarily ready yet, we may need to add a constant, c. So pick a line, the easiest seems to be 1 1 1 1 1 = 5. Now, since we know the five multipliers, we can say that 1x3 + 1x1 + 1x4 + 1x1 + 1x5 + c = 5 5 + c = 5 c = 0 The constant is 0. So just multiplying each digit by its respective multiplier gives the correct answer. You still need to verify your formula by calculating the right hand side of a few lines.