Explorations In Math

Can you solve these Puzzles?

Published on September 26 2011

There is so much to enjoy about math! I’ve written about some of these in “Math Matters” over the summer: math games (whether commercial or otherwise), going to any of the math/science museums in our area (KidsQuest, Pacific Science Center, Museum of Flight), looking for patterns in both numbers and nature, and finding great math websites. Here’s another: math puzzles (or brain teasers as they’re sometimes called). There are two kinds of math puzzles. One kind involves no math but it does require the kind of flexible thinking and logical reasoning that solving math problems requires. Here’s an example of such a puzzle:

What number goes in the box with the question mark? Why?

Here’s a puzzle that does require math in addition to flexible thinking and logical reasoning:

A man was condemned to death by the king but told the man he would give him a 50% chance of saving his life. The king had two large urns brought into the throne room. “Prisoner,” the king said, “in one urn are 99 red balls and 1 white one. In the other, 99 white ones and 1 red one. You will be blindfolded and be placed between the two urns. Choose one of them and remove one ball from it. If it is red, you die; if it is white, you live.”

The prisoner thought. “If I pick the urn with 99 white balls, I still have 1 chance in 100 of getting the red one and dying, and if I pick from the other urn, I still have 1 chance in 100 of  living. Overall, my odds are 50/50.”

He then asked the king, “Sire, may I rearrange the balls in the urns to improve my chances?”

“As long as neither urn is empty, you may do as you please,” the king replied.

Quickly, the prisoner rearranged the balls so that his odds of living increased from 50% to almost 75%. What did he do?

I have given both these puzzles to students I work with and both actively engaged them. They refused to give up and didn’t want to be told the answer, even when we were running out of time. I’ll give you a week to solve them before I give the answers. Send your solutions to davega@eimath.org. The first person to solve both correctly will receive a math game. Enjoy!

Dave Gardner

Mathematician in Residence


3 Reader Comments to “Can you solve these Puzzles?”
  • By St. George 4th Graders on October 12, 2011 | Reply

    We solved your first puzzle and the answer is 4 because that box connects to 4 other boxes. The box labelled “1″ connects to only one other box and the box labelled “2″ connects to 2 other boxes.

    For the second puzzle, our guesses are:
    1. Perhaps he rearranged the balls so that there were three white and one red in each and he asked to get rid of the rest since the King said the urns just had to not be empty.

    2. Perhaps he rearranged the urns so that there was only one ball in one urn and all the other balls were in the other urn.

    3. Perhaps the prisoner rearranged the urns to have only white in one and only red in the other and then he picked from the white urn.

    4. He rearranged them so that their was fifty of each color in each urn.

  • By Dave Gardner on October 12, 2011 | Reply

    Great work St. George 4th graders. Your mathematical thinking is awesome!
    Your answer to #1 is perfect and your second response to #2 is almost spot-on. You simply neglected to state which color ball goes by itself in one urn. Which one is it? You’ll be getting your classroom math game soon! Congratulations!

    Mr. G.

  • By St. George 4th Graders on October 13, 2011 | Reply

    He had one white ball in one urn and the rest in the other urn. This way, he would have nearly a 50/50% chance if he reached into the urn with several balls and 100% of living if he picked the urn with the one white ball. This problem was tougher for us than the line puzzle!

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