492 Cubed is Sum of 3 Positive Cubes in 13 Ways/Mistake

Source Work

 * $6$: Number Recreations
 * Narcissistic Numbers:
 * Miscellaneous
 * Miscellaneous

Mistake

 * ... an integer less than $1,000$ whose cube could be represented in $5$ distinct ways as the sum of the cubes of $3$ positive integers...
 * In the table below, three solutions to the problem are shown that have gone far beyond ...


 * $\begin {array} {ccc}

& n = 492 & \\ a & b & c \\ 24 & 204 & 480 \\ 48 & 85 & 491 \\ 72 & 384 & 396 \\ 113 & 264 & 463 \\ 144 & 360 & 414 \\ 176 & 204 & 472 \\ 207 & 297 & 438 \\ 226 & 332 & 414 \\ 246 & 328 & 410 \\ 281 & 322 & 399 \\ \end {array}$

Correction
The $5$th triple is wrong.

The first number is $114$, not $144$.