A baker’s dozen (thirteen) bagels have the property that any twelve of them can be split into two piles of six each, which balance perfectly on the scale. Prove that all the bagels have the same weight.

Nobody asked.

seems pretty obvious to prove tbh
if there is 6 on each side of the scale plus one leftover
then for the property to hold true, the leftover must have the same weigh as every other bagel
if that wasnt the case, if you swapped any bagel with the leftover, the scale would tip, meaning the property is false
and since the property must be true for any 12 bagels on the scale, its the same as saying it must be true for any one bagel left over
therefore all of them must have the same weigh

P16097
Looks good to me.