How many times can you spot the letter "l" in:
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.
From here on, it's easy.
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.  Wishsful thinking with a PR twist! . . . PAP!
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."
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.
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).
Should you switch or stay?
Puzzle built by John and Jeanine Denoma.
Da Vinci Days, 2002 Use "Back" to return to the cubes page.

user
I found one hundred and thirteen "l"s in the provided passage.
assistant