Da Vinci Days
2002
Oregonians for Rationality booth

The Cantankerous Cube
five by five by five

You probably start solving this puzzle by filling the lowest layer with blocks.  When you have just one more piece to add to the layer—assuming that you haven't yet used any of the little "stick" pieces (1X1X3)—you will discover that a stick piece is the only kind of piece that can fill up the layer. 

From this point on you can guarantee frustration, failure and fultility in a very simple way:  just put two of those sticks so that they lie in the same plane,  Any plane.  There are fifteen planes in the finished 5X5X5 cube.  Five are horizontal, like the one illustrated below at the bottom of the puzzle.  And five stand up vertically running front to back and another five stand up vertically running left to right.
 


Puzzle pieces by John Denoma.

The problem is that all the other pieces add an even number of little 1X1X1 units, and every plane has 25 of those units, an odd number.  Only the sticks can add an odd number.  Each stick will add an odd number to five different planes: it will add one unit in three different planes and three units to two planes.  So there's just enough stick pieces to go around with none to spare.  Put two into any one plane, and you won't have enough to go around.
Most solutions (there are many) will have the three stick pieces arranged as though they were connected at their corners to make a three-piece snake.  And we can expect the snake to run from one corner to the opposite corner.

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Here are three stick pieces taken from the one puzzle of the six similar puzzles that has those pieces connected into a snake.   ¯¯¯>
 
 
DETAILS of puzzles
From here on, it's easy.
A more graphic explanation from Da Vinci Days, 2003
 

Here,
a little math goes a long way.

 
Let's go a little further.

Statistics is seldom understood very well.  In fact, statistics is most often seen as a pack of lies told us by people who think like advertisers and political campaign managers.  Pick and choose your stats and you can "prove" any damn thing you wish.  Wishsful thinking with a PR twist! . . . PAP!

The scientific truth lies in a completely different dimension:  statistics is one of the most powerful tools in the scientist's toolbox.  And any scientist who picks and chooses his or her data to prove a point will soon be doing something else for a living...perhaps PR for a politician?  (This isn't a precaution for just the would-be scientist: everybody should become able to handle a little statistical reasoning.  It is, for example, important in economics: in the stock market, the state lotteries, the gambling casinos, in ordering manufacturing materials...everywhere.)

We might consider the odds of getting that pesky 5X5X5 cube together.  If you don't know about distributing those stick pieces with perfect frugality, what are the odds of your putting two of them into one plane.  Almost a certainty, of course.  Your odds of success, then are about the same as your odds of winning the big one in a state lottery: "ZERO, to eight significant figures."

Bob Park's phrase. See What's New
At Da Vinci Days, O4R demonstrated something that, several years ago, became an embarrassment to a lot of professional mathematicians who demonstrated that they didn't understand the logic.  It's a puzzle that Marylin vos Savant once put in her Parade column.  Monty Hall once used it, too.

The puzzle experiment was run using random placement of the prize—randomness is one of the statistical concepts that needs better public understanding  The results of two days of people answering that puzzle were tabulated with hazel nuts in clear-walled columns.  Seeing the running tabulation gave a clear answer to the problem, even if you couldn't work it out logically for yourself. 

Look at the results.  Do you think some other kind of nuts would have helped people see the "obvious"?
,,,
.................................win      lose                  win      lose
 .................................107       48                   94       170
 It's clear that those who chose "Switch" were far more likely to win than those who chose "Stay."

And those who played had a strong bias toward "Stay"  (that is, LOSE).

The problem:

A hustler presents you with three boxes into one of which he places a valuable prize, but you don't know which one.  You get to choose one box and keep the prize if it's there.  You choose, but he doesn't tell you if you won.  Instead, he says to you "I'm going to open one of the boxes and show you what's inside."  He does (and you see that there's nothing inside), and then says, "Now, would you like to stay with your first choice, or would you like to switch."

Should you switch or stay?
Puzzle built by John and Jeanine Denoma.
Hear Kieth Devlin dicuss this on NPR
(May 24, 2003: go to "Probability")

 
 
 
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