HMMT 二月 2000 · ADV 赛 · 第 5 题

求所有等于各位数字立方和的三位数。
四个 3 位 Armstrong 数：$1^{3}+5^{3}+3^{3}=153$，$3^{3}+7^{3}+0^{3}=370$，$3^{3}+7^{3}+1^{3}=371$，$4^{3}+0^{3}+7^{3}=407$。
$\boxed{153, 370, 371, 407}$。

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英文解析

Find all 3-digit numbers equal to the sum of the cubes of their digits.
Let $n = 100 a + 10 b + c$ where $a \in \{1,\ldots,9\}$, $b,c \in \{0,\ldots,9\}$. Need $100 a + 10 b + c = a^{3} + b^{3} + c^{3}$.
Check each digit combination:

• $153$: $1^{3} + 5^{3} + 3^{3} = 1 + 125 + 27 = 153$ ✓
• $370$: $27 + 343 + 0 = 370$ ✓
• $371$: $27 + 343 + 1 = 371$ ✓
• $407$: $64 + 0 + 343 = 407$ ✓
  These are the four 3-digit Armstrong (narcissistic) numbers.
  $\boxed{153, 370, 371, 407}$.