"Eurekas" are sometimes insights which might find use anywhere.  They are like perceptions, "perceptions of the abstract."  These Web pages explore these "perceptions at the edges of human comprehension."  Puzzles that seem "tricky" and have points that seem to evade your notice often give clues to the nature of that elusive edge of comprehension.
Some puzzles seem impenetrable at first, then suddenly:
You see it!


You don't see it gradually; you don't see it in little steps; you just look at it in some different way, and there it is.

A carpenter, working with a buzz saw, wishes to cut a wooden cube, three inches on a side, into 27 one-inch cubes.  He can do this job easily by making six cuts through the cube, keeping the pieces together in the cube shape.  Can he reduce the number of necessary cuts by rearranging the pieces after each cut?
up or down?...click here

What is the direction, up or down, of the acceleration of a freely bouncing ball at the bottommost point of its bounce, that is, at the instant its velocity changes from down to up?

Start with a chess board and 32 dominoes.  Each domino is of such size that exactly covers two adjacent squares on the board.  The 32 dominoes therefore can cover all 64 of the chess board squares.  But now suppose we cut off two squares at diagonally opposite corners of the board and discard one of the dominoes.  Is it possible to place the 31 dominoes on the board so that all the remaining 62 squares are covered?  If not prove it impossible.
Start with four figures: a red square, a blue  triangle, a red triangle, and a blue square. Someone says, "I am thinking about one of the colors and one of the shapes, and if a figure has either the color or the shape I'm thinking about (or both) then it is accepted.  Otherwise it is rejected.  I accept the red triangle."  The problem is: May any of the remaining figures be consistently rejected? 

In a set of cards each card has a number on one side and a letter on the other.  Four cards are lying on a table.  They show an "I", an "N", a "6", and a "3".  Someone suggests the hypothesis: If a card has a vowel on one side then it has an odd  number on the other side.  The problem is to determine which cards must be turned over to test the hypothesis.  No card is to be turned over unless necessary to test the hypothesis.

  I & 6
Five cards are placed in a row as shown.  All card backs are either marble or checker-patterned.  Are all the cards with marble backs aces?  The problem is not to answer the question but to determine the minimum number of cards that must be turned over in order to answer it. 
You are on a quiz show and the quizmaster shows you three doors.  Behind one door is the winning lottery ticket, worth tens of millions.  Behind the other two doors are copies of the 95-page "instruction" booklet that came with the first version of Windows 95.  You are to pick a door, and you get the prize which is there.  You choose a door, but now the quizmaster opens, not the door you chose, but rather one of the other doors.  There is a Windows 95 booklet.  Now, the quizmaster gives you the option of sticking with the door you chose, or picking the other door, both of which are still closed.  Should you switch or stick?

3: 1st, 3rd, & 4th

...Are you sure?...Really sure??...THINK AGAIN!!


You paddle your kayak up the river from your camp to fetch your camera which you left on a rock upstream a bit.  The river flows at a uniform 2 mi/hr.  You paddle (on still water) at a uniform 3 mi/hr.  It takes 30 minutes to reach your camera.  If you paddle all the way back to your camp, how long will the return trip take?
"I could care less."  -- "I couldn't care less."
"The record low is minus 90°F" -- "The record low is minus 90°F below zero."

What do these statements say? --  Why do some people use the wrong one?
These are unusual puzzles.  They are extremely simple.  Most of them, however, have answers that evade detection.  The puzzles are simple but difficult.  They require "perceptions" of things at (or perhaps a little outside) the edges of human comprehension.  The buzz-saw and checkerboard answers are inside the edge for most everyone.  So these might be used as exemplar for the more difficult of "perceptions", the perception of the abstract.  When you see, you know with

buzz-saw certainty.
If you can see it, you find it hard to believe that others don't.
If you don't see it, you find it hard to believe that anything is there is there to see.

Click HERE for a set of puzzles used to help teach the concepts of information content and entropy.
Quantum Jumps
Quantum Jumps
to Opening Page
to Index Page
to the simple but difficult puzzles
to the Misconceptions workshop
to Wason's gate into the Edge
to the fine print
to Second Glances Navigator
to the zoo and the perceptions exemplar
to The Platinum Plover Egg
to Accurate Maps
to Glen Canyon Navigator
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