What is missing? Range of integers from 1 to 21 cubed, some missing, sum to a cubed integer.

Number Theory Level 2

This problem's question: Which three integers from 1 to 21 are missing when the remaining integers are cubed and then summed and the result is itself a cube? Write the answer as the concatenation of the three integers from smallest to largest placed left to right. For example, if the missing integers were (and they are not) 2, 15 & 21, then the answer is 21521.


The answer is 41217.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Plus@@ { 1 , 2 , 3 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 13 , 14 , 15 , 16 , 18 , 19 , 20 , 21 } 3 3 ⇒ 3 6 3 \sqrt[3]{\text{Plus}\text{@@}\{1,2,3,5,6,7,8,9,10,11,13,14,15,16,18,19,20,21\}^3} \Rightarrow 36^3

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brilliant solution!

A Former Brilliant Member - 2 years ago

Is the answer unique?

Pi Han Goh - 2 years ago

With the range of possible integers being from 1 to 21 and with three integers missing, yes. Checked by exhaustive search. There are other solutions with different numbers of integers missing.

A Former Brilliant Member - 2 years ago

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Do you expect us to manually evaluate ( 21 3 ) = 1330 \binom{21}3 = 1330 possible combinations by hand?

Pi Han Goh - 2 years ago

With the range of possible integers being from 1 to 21 and with three integers missing, yes.

A Former Brilliant Member - 2 years ago

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