What is wrong with the following proof that all cupcakes hav

What is wrong with the following ‘proof’ that all cupcakes have the same flavor?

“Suppose there are n cupcakes in the world. We induct on n. If n = 1 the result is clear (since any cupcake has the same flavor as itself). Fix a positive integer n > 1 and inductively assume that any collection of n 1 cupcakes has the same flavor. We show that all n of the cupcakes in the world have the same flavor. To do this, place the cupcakes in a line and label them c1, c2, . . . , cn. By the inductive hypothesis, the first n 1 cupcakes c1, c2, . . . , cn1 and the last n 1 cupcakes c2, c3, . . . , cn all have the same flavor. In particular, the cupcake c1 has the same flavor as the middle cupcakes c2, c3, . . . , cn1. Also, the cupcake cn has the same flavor as the middle cupcakes c2, c3, . . . , cn1. We conclude that all of the cupcakes c1, c2, . . . , cn1, cn have the same flavor, as desired.”

Solution

we cannot go from n=1 to n=2 using induction.

The problem is when n=2

we are assuming c_1,c_2....c_n-1 and c_2,c_3,.....c_n has something common

but when n=2 these are two seperate sets

c_1 and c_2

they don\'t have anything common

so their flavour can be different