Sanderson M. Smith

CHRISTMAS SEASON PUZZLE

The father of one of my students sent me the following problem. I have jazzed it up a bit to make it fit the season, but have not changed the logic required to solve the problem.

===================================================================

Santa has a collection of ornaments, all identical except for color. Some are red and some are green.

A bag labeled RED contains only red ornaments.

Another bag labeled GREEN contains only green ornaments.

A third bag labeled MIXED contains at least one ornament of each color.

Santa's elves are continually seeking ways to keep Santa mentally sharp. (After all, he is an old guy.) So they intentionally switch the labels on the three bags so that each bag is incorrectly labeled.

The elves challenge Santa to get the labels back on the proper bags. But he must do it in the following way: Without looking into any of the bags, he can reach into any one of them and withdraw an ornament and note its color. Santa can continue to withdraw ornaments from any bag, one by one, and inspect them until he can correctly label each bag.

WHAT IS THE FEWEST NUMBER OF ORNAMENTS SANTA MUST INSPECT IN ORDER TO CORRECTLY LABEL EACH BAG?

============================
(The correct answer appears below. You must scroll down to see it. You might want to see if you can reach a conclusion before peeking.)

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

Santa need only inspect one ornament. Here is what he could do.

Since the bags are incorrectly labeled, the bag marked MIXED contains ornaments of only one color.

Pick an ornament from the incorrectly-labeled MIXED bag.

* If the selected ornament is RED, then that bag should be labeled RED.

Now, the bag that is incorrectly labeled GREEN must be the MIXED or the RED.

But, it can't be RED, since that bag has already been identified. Hence it must be the MIXED bag.

Finally, the bag that is incorrectly labeled RED must be the GREEN bag (the only remaining possibility).

==========
* If the selected ornament is GREEN, a similar argument can be presented. (Just interchange the words RED and GREEN in the above presentation.)
==========