100 numbers were written around a circle. The sum of the 100 numbers is equal to 100 and the sum of six consecutive numbers is always less than or equal to 6. The first number is 6. Find all the numbers.
Let S be the sum of 6 consecutive numbers on the circle. S<=6
The sum of 96 consecutive numbers in the circle is the sum of 16 blocks of 6 numbers. Then the sum of 96 numbers is 16*S. Then 16*S<=96.
The sum of all 100 numbers is 100. So then the four numbers not in the group of 96 numbers has a sum of 100-16S. Then 100-16S>=100-96=4.
The partition of the set of 96 vs set of 4 is arbitrary. Therefore the sum of four consecutive numbers in the circle is always greater than or equal to 4.
A circle of 100 numbers can be partitioned into 25 such sets of 4. Then the sum of all 100 numbers must be greater than or equal to 4*25=100.
However we are told that the sum of all 100 numbers is exactly 100. This then means the sum of a set of four numbers is exactly 4. Backtracking further then we must have 100-16*S=4, which means S=6 so then the sum of a set of six numbers is exactly 6.
Given any set of 6 consecutive numbers, it can be partitioned into a set of four numbers and an adjacent pair. The adjacent pair must have a sum of 6-4=2.
If one number in an adjacent pair is 6 then the other is 2-6=-4. So starting from the given 6 it must be that the pattern around the circle alternates 6, -4, 6, -4, etc.
Solution
