Da Vinci Problem

Number Theory Level 2

Imagine that you are in a cave that the only way to get out is with a code which gives you two patterns of the code to get the next two and get out of the cave in which the evil Da Vinci left you. {1210}, {3211000}.

{9310001001}, {18410000001001} {122100010}, {72100000100} {9310001000}, {62100000010000} {6210001000}, {9210000001000} {521100100}, {7210100001000}

Elijah Frank by Elijah Frank , 14, Ecuador

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2 solutions

Vishruth K
Apr 7, 2021

The two numbers you were given are autobiographical numbers, and two others are only mentioned in {6210001000}, {9210000001000} . So that's the answer.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Elijah Frank
Mar 17, 2021

the pattern seems a little bit tricky but have a property named self-referential numbers which it's going like this: {1210} 1 zero, 2 ones, 1 two, 0 three. {3211000} 3 zeros, 2 ones, 1 two, 1 three, 0 four, 0 five, 0 six. Then doing the same we obtain the two codes with this simple method: we known that the next is 6 so it's 6 zeros and we need 2 ones, 1 to represent that two and the other to represent six: {6210001000} and the final zero tell us that the next pattern is a 9 so we need 9 zeros, also the 2 ones for the same purpose: {9210000001000}. now we can escape from the cave.

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These numbers are called autobiographical numbers. You can practice them here: https://brilliant.org/practice/autobiographical-numbers/?p=1

Avner Lim - 2 months, 3 weeks ago

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