Brain teaser

Posted by I. S. on Jun 18, 2002 at 14:32
(130.207.83.108)

Everyone in the U.S. has the same age! Proved by induction:

Step 1: In any group that consists of just one person, everybody in the group has the same age, because after all there is only one person!

Step 2: Therefore, statement S(1) is true.

Step 3: The next stage in the induction argument is to prove that, whenever S(n) is true for one number (say n=k), it is also true for the next number (that is, n = k+1).

Step 4: We can do this by (1) assuming that, in every group of k people, everyone has the same age; then (2) deducing from it that, in every group of k+1 people, everyone has the same age.

Step 5: Let G be an arbitrary group of k+1 people; we just need to show that every member of G has the same age.

Step 6: To do this, we just need to show that, if P and Q are any members of G, then they have the same age.

Step 7: Consider everybody in G except P. These people form a group of k people, so they must all have the same age (since we are assuming that, in any group of k people, everyone has the same age).

Step 8: Consider everybody in G except Q. Again, they form a group of k people, so they must all have the same age.

Step 9: Let R be someone else in G other than P or Q.

Step 10: Since Q and R each belong to the group considered in step 7, they are the same age.

Step 11: Since P and R each belong to the group considered in step 8, they are the same age.

Step 12: Since Q and R are the same age, and P and R are the same age, it follows that P and Q are the same age.

Step 13: We have now seen that, if we consider any two people P and Q in G, they have the same age. It follows that everyone in G has the same age.

Step 14: The proof is now complete: we have shown that the statement is true for n=1, and we have shown that whenever it is true for n=k it is also true for n=k+1, so by induction it is true for all n.