“Een, vier and negen are the Dutch for 1, 4, and 9; so it is appropriate that I can make the following statement:

EEN, VIER, and NEGEN are perfect squares.

In this statement digits have been consistently replaced by capital letters, different letters being used for different digits. No number starts with a zero. Find the numerical value of the square root of (EEN * VIER * NEGEN).”

Okay, this sounds very complicated. And that’s what I thought at first, but then I looked at the first number, EEN. It’s a three-digit square number, and the hundreds digit and the tens digit were the same. There are only two three-digit numbers with that property: 225 and 441. That means EN is either 25 or 41. Do you see it? DO YOU SEE IT?

Yes. NEGEN. It’s a palindromic square number, and it either ends in 25 or 41. That means it is either 52_25 or 14_41. The easier case was if EN = 25. Since the only way that can be true is if the square root is a multiple of 5. You see, for 10a + 5, the square of it is 100a(a+1) + 25. In other words, the last two digits are always 25, and the beginning digits before that are a(a+1), where the square root is 10a + 5. Since 50000 > 40000 = 200^2, which meant that if EN = 25, then NEGEN was around 235^2. Testing values, I found that 235^2 = 55225, which is pretty close, but not enough. Any higher, and it would be already too much. For 225^2 = 50625, too low. Since it has to be a perfect square, EN cannot be 25. Therefore, EN = 41.

Alright! We got somewhere. EEN = 441, NEGEN = 14_41. The only square for that case is 121^2 = 14641.

The final case, and the harder part. VIER.

You remember from the last post, “CAST THE ODD ONE OUT“? I listed every four-digit square numbers, and Since E = 4, the square is __4_. The digits are all unique, and with the tens digit being 4, the only solution I could find was 3249. We did it!

Never mind. We didn’t do it yet. We need to solve the actual problem: the square root of the product. Easy enough; 441 = 21^2, 14641 = 121^2, and 3249 = 57^2. 21 * 121 * 57 = 144837, and that’s the solution.