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With a bit of extra help from assistant: Otherwise EDITED: Now just for fun:

The only way to prove is to make it, which is very tedious. It is trivial to show it might possibly exist or not - we require at least N 1s, M 2s etc to build all numbers with that digit. If we cons …

There is a third possible interpretation of the question - the largest possible score with particular subscores that guarantees the finals. This is the highest possible score that gets a candidate to …

Some hand analysis first. This is sufficient to prove there are infinitely many 4 element sets. These are NOT all of the 4 element options. I believe you can start with any number (at all), then pic …

Well, what we need is floor of 2017 * ln(2)/ln(3), which will be 1272. Now the challenge is essentially getting ln(2)/ln(3). For that, we can use integral. ln(2) = integral from 1 to 2 of (1/x dx). ln …

Initial guess: Math to the rescue: Lateral thinking:

No need for computers, hand solving will do (though I did use code in the end to avoid making any calculation mess). It is trivial to see that you should start from the end. Then preferentially remov …

As we note, after a round of both losing, everything is the way it was initially. Let's proclaim chance to draw a red ball is r, chance to draw a blue ball is b = 1-r. Probability A wins in a round is …

Well, mean is infinite (easy enough to see): Median is finite though. In code I slightly simplified the problem - function [n,m,k] = so_coins(n) % n=number of coins left in the urn. m=0; % number o …

As all the reasonable solutions seem to be disallowed: It is trivial to get to 4.8l, simply put 8l container inside the 6l one and fill the 6l one to the edge so it holds 4.8l. There is no other stack …

220 is 2^2 * 5 * 11, it can be obtained as (2,10,11) and (4,5,11) 216 is 2^3 * 3^3, it can be obtained as (2,9,12), (3,6,12), (3,8,9), (4,6,9), (2,3,4,9) 224 is 2^5 * 7, it can be obtained as (14,16), …

Inspired by Two dice with the same probability for each sum? To cheat in a game of sums, you get yourself a pair of magic dice. That pair behaves in a wonderful way where each individual die is fair …

Non-programming solution to find the number:

Human mind can brute force too :P 2 is obvious and 3 is easy. Then: List of prime distances from 3 onward: 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, (5) - to get from 61 to 2. Draw path in co …

Each step increases sum by the 2*(x+y). It is obvious that the minimum increase of sum in a particular step is if you take the lowest two numbers available. But this is not quite enough to show greedy …