Question

Hard Maths Olympiad Question Here. Feel free to give it a go

Answer 1

I have several identical coins placed on a table. I play a game consisting of one or more moves. Each move consists of flipping over some of the coins. The number flipped stays the same for each game, but may change from game to game. A coin showing 'heads' is represented by H and tails is represented by T. The same coin may be selected on different moves. In each game, we start with all coins showing tails.
a) starting with 14 coins, flipping over 4 coins on each move, explain how to finish with exactly 10 coins heads up in three moves
b) Starting with 154 coins and flipping over a fixed odd number of coins on each move, explain why i cannot have all 154 coins heads up at the end of three moves.
Enjoy!

Answer 2

@dan815 @Kainui @nincompoop @rational

Answer 3

for part a we may try like this

Initial :
TTTT TTTT TTTT TT

Move1 :
HHHH TTTT TTTT TT

Move2 :
HHHH HHHH TTTT TT

Move3 :
HHHH HHHT HHHT TT

Answer 4

ok the second part is where the struggle begins lol

Answer 5

for part2 we want to solve below in nonnegative integers
\[(2n+1) + (a-b) + (c-d) = 154\\

a+b = c+d = 2n+1
\]

Answer 6

@rational can u specify what all the pronumerasl are