The main hint is that the left vertical number is the same as the right vertical number.

Then, consider which twice perfect cubes (3-digit ones) have equal first and last digits: `6 8 6 = 2 * 7 ^ 3 ` only. You can test all others by use of...

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The main hint is that the left vertical number is the same as the right vertical number.

Then, consider which twice perfect cubes (3-digit ones) have equal first and last digits: `6 8 6 = 2 * 7 ^ 3 ` only. You can test all others by use of a calculator.

Further, which even squares fit: `4 8 4 = 2 2 ^ 2 ` and `6 7 6 = 2 6 ^ 2 .`

The only multiple of `3 7` which has equal first and last digits AND the middle digit 1 is `1 1 1 = 3 * 3 7 . ` The middle digit must be 1 because the middle vertical number must be less than `2 0 0 .`

This way, we need to choose which even square to use. The middle vertical number is `1 7 8 ` or `1 8 8 ` and it must be a multiple of 47. It is `1 8 8 = 47 * 4 .`

The final square will be as follows:

1 1 1

4 8 4

6 8 6.