A 23" x 23" square metal plate needs to be fixed by a carpenter
onto a wooden board. The carpenter uses nails all along the
edges of the square such that there are 24 nails on each side
of the square. Each nail is at the same distance from the
neighboring nails. How many nails does the carpenter use?

If I understand your question, do we really care about the dimensions of the square?

If all we're doing is making sure we have 24 nails on all four sides, we put 24 nails at the top, 24 nails at the bottom, and 22 nails on the left and right. Now they're evenly spaced and that makes a total of 92 nails.

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Another way to do this is to note that 4(24)= 96. However, the 4 nails at the cornes are counted twice: for example the nail at the upper left corner is counted as one of the 24 along the top and once as one of the 24 along the left side. To account for that double counting we subtract off 4 nails: 96- 4= 92 nails masters said.