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?

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 c_1, c_2,..., c_n. By the inductive hypothesis, the first n - 1 cupcakes c_1, c_2,..., c_n - 1 and the last n - 1 cupcakes c_2, c_3,..., c_n all have the same flavor. In particular, the cupcake c_1 has the same flavor as the middle cupcakes c_2, c_3,..., c_n - 1. Also, the cupcake c_n has the same flavor as the middle cupcakes c_2, c_3,..., c_n - 1. We conclude that all of the cupcakes c_1, c_2,..., c_n - 1, C_n have the same flavor, as desired.\"

Solution

The statement, the last n-1 cupcakes have the same flavour(c2,c3...cn), is wrong. In mathematical induction, provided the n-1 th expression is true the nth expression has to be proved. In this case two assumptions are made. Hence, this proof is wrong.