We need to arrange the following numbers in a circular shape (1,2,3,4,5,6,7,8,9,10) with the following conditions:
for each one we should prove that it's impossible according to the Pigeonhole principle
i) The sum of three numbers next to each other is no less than 17
ii) The sum of three numbers next to each other is no less than 18
iii) The sum of three numbers next to each other is no less than 19
I tried to solve this using beads, I said that we have 55 beads (the sum of all numbers), at first we need to put one beed on every chair around the table since 0 is not allowed then we are left with 45 beads, one chair will have 10 beads for sure so we need to add to a random chair 9 more beads so we are left with 36 beads.
But I can't continue from here...