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I am conscious of the fact that this question is very elementary, but I have no apparent idea as to how one intends to go about it. If someone could kindly provide a layman possessing the soul of myself some mathematical intuition, it would be appreciated greatly.

Consider the following counting game, in which Person A will say the number 1, then Person B states the next two numbers (i.e. 2 and 3), Person C shall say the next three numbers, prior to the moment when Person A states the next four numbers. This process continues to proceed in a rotation, until the number googleplexian is obtained. The question I wish to pose is: Out of Person A, B and C, who states the last number?

Arthur
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LÜHECCHEgon
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    Hint: The total number of numbers spoken by the trio after $n$ terms is the $n^{th}$ triangular number, $\frac {n(n+1)}2$. though, of course, the extreme size of the number involved might make the computation difficult. – lulu Jul 30 '19 at 11:05
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    yes if you can solve for n(n+1)/2 = googolplex, then divide it by 3 and the modulo number says it. – user29418 Jul 30 '19 at 11:07
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    Hint: $100\equiv1\mod3\implies10^{100}\equiv1^{100}=1\mod3$ et cetera. – drhab Jul 30 '19 at 11:09
  • @user29418 What's the remainder mod $3$ of the sum of the first half a googol digits of $\sqrt{2}$? – Magma Jul 30 '19 at 11:26
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    @Magma you know very well that it's not possible to do a clever manipulation. your question is ill-posed. – user29418 Jul 31 '19 at 16:00

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