Two babies A and B are playing a game with $2022$ bottles of milk. Each bottle has a maximum capacity of $200$ml, and initially each bottle holds $30$ml of milk. Starting from A, they take turns and do one of the following: (1) Pick a bottle with at least $100$ml of milk, and drink half of it. (2) Pick two bottles with less than $100$ml of milk, pour the milk of one bottle into the other one, and toss away the empty bottle. Whoever cannot do any operations loses the game. Who has a winning strategy? Proposed by Chu-Lan Kao and usjl
Problem
Source: 2022 Taiwan Mathematics Olympiad
Tags: combinatorics, Taiwan
USJL
19.02.2022 12:03
A proposer remark: One other version of this is that we start with $1011$ bottles with $90$ml of milk, and $1011$ bottles with $150$ml of milk. The solution we have for this version is completely different from the one for the version in the contest. One interesting question would thus be: determine the person with a winning strategy given any initial configuration. This seems to be a really hard problem, and we would be interested in any partial progress toward this direction.
fffff
25.02.2022 20:08
I think we can make it easier if we simply consider that both A and B can ensure that there's only 120ml when some bottles have milk more than 100ml.This is because 2022 is an even number.Whenever 90ml appears,A or B can make it into 120ml for there is at least one bottle have 30ml. (Of course, in a limited time ,this game stops.) Now let the number of bottles which have 60ml,90ml,120ml be written as X,Y,Z. Meanwhile, let the number of we use(1),(2)be written as A,B. A-2B=X+2Y+3Z Now A+B is congruent to X+Z+B(module 2),however, at the end of this game, X=1,Z=0,B=(2022*30-60)/60 is an even number. So A+B eventually equals to an odd number. This means that now B can't continue.So A can win easily. I think that when the number of bottles given at the beginning is congruent to 2 (module 4),A will win.When it congruent to 0 (module 4),B will win.
andywu
27.02.2022 09:48
https://hackmd.io/@sine/TMO_2022P4
R-long
27.02.2022 11:36
@above I think actually you don't need to write (2). Because you say you make (1) first, part 1 was finish when A have to do (2). It's impossible to occur.