The Online Language Learning Database
OPTICAL ILLUSIONS

When you think you have an answer to one of the puzzles below, e-mail it to and we'll e-mail you back the solution.

DARMO’S PUZZLE

Number the eight areas with 1 through 8 so that consecutive numbers are not adjacent.

 

LEIBNIZ’S PUZZLE

Of the seven pieces on the left, discard one, and rearrange the remaining
six in the pattern shown on the right. Is there anything wrong here?

 

ACE'S PUZZLE

With three straight cuts, divide a washer into nine pieces. 
Leave the pieces in place after each cut.
(The cuts above form only seven pieces.)

 

DUNCAN’S PUZZLE

With three straight cuts (they can be at an angle),
 divide a doughnut into thirteen pieces. 
Leave the pieces in place after each cut.

 

MY SISTER’S PUZZLE

My sister is three times as old as I was ten years ago, and she’s twice as old
 as I was when she was my present age. How old is my sister?  How old am I?
Check to see if she's now three times my age ten years ago.  
How old was I when she was my age now?  Is she twice that?

 

SHADANIAN’S PUZZLE

Cut a 6 12 carpet into two pieces so they can be sewn together into an 8 9 carpet.

 

HOLZHAUER’S PUZZLE

Cut an 8 8 27-inch wood beam into four pieces
so they can be reassembled into a 12-inch cube.

 

TOM & RAY’S PUZZLE

Can you cover the checkerboard with 32 dominoes? Can you
cover the truncated checkerboard with 31 dominoes?

 

SHEARSON’S PUZZLE

Intertwine a cord with a pair of scissors as shown. With
ends X and Y anchored, remove the cord from the scissors.

 

LOUIS’S PUZZLE

Louis lives four blocks north and four blocks west of school. How many
different routes can he take to school (always heading south or east)?

 

MATT'S PUZZLE

Arrange eight queens on the chess board so no queen is
attacking another. The arrangement shown above almost
works; two queens still attack each other diagonally.

 

TERRY’S PUZZLE

Tear and fold your business card so it looks like this.

 

RUSS’S PUZZLE

For a sequence of three 1s and 0s, there are eight possible patterns:

000
001
010
011
100
101
110
111

For a sequence of n 1s and 0s, there are 2n possible patterns.

For a sequence of three 1s and 0s with no adjacent 1s, there are five possible patterns:

000
001
010
100
101

For a sequence of n 1s and 0s with no adjacent 1s, how many possible patterns are there? (Try for a few values of n, see a pattern, and generalize. Then prove.)

 

ARNIE’S PUZZLE

Arnie has opened a 12-oz can of soda in the car. (The can itself weighs 1/2 oz.)
If he drinks nothing, the center of gravity is at the center of the can. If he drinks
it all, the center of gravity is again at the center of the can. How much soda should
Arnie drink so the can has the least chance of tipping over if he comes to a sudden
stop? (Assume a fluid ounce weighs an ounce.)

 

HOWARD’S PUZZLE

 

A stick is broken at random in two places. What
are the odds that you can form a triangle of the pieces?

 

LIBRO’S PUZZLE

What is the minimum number of books in a skewed stack so the
top book projects one book width beyond the bottom book?
How many so it projects two book widths?

 

BINNY'S PUZZLE

Pick four weights so some combination of them in the left pan
will balance a sack of wheat from 1 to 15 lb. by 1-lb. increments.

 

TRYON'S PUZZLE

Pick four weights so some combination of them in the left and right
pans will balance a sack of wheat from 1 to 40 lb. by 1-lb. increments.

 

JIMMY'S PUZZLE

Jimmy, in a west-coast state, calls his mother in an east-coast state. After
a while she says she really should be getting to bed, and mentions the time.
"Hey!" says Jimmy, "it's the same time here!" What states are they in?
(There is no trick in the wording here. A "west-coast state" is Washington, Oregon, or
California, and an "east-coast state" is one from Maine to Florida. "The same time" means
what the clock reads in that time zone when correctly set to standard or daylight-saving time,
according to the time of year.)

(USA time zones)

 

GRENZA'S PUZZLE

A planet with no bodies of water is divided into countries by lines that meet (terminate) at points. No country is an "island" (surrounded by one country). There are 37 lines and 15 points. How many countries are there?

 

PIERRE'S PUZZLE

Pierre has thrown several pieces of 24 lumber into the lake. Some pieces float with the 4" side up and some with the 2" side up. What is the deciding factor?

 

RACHEL'S PUZZLE

There is an infinite number of circles. What fraction
of the area within the triangle do the circles occupy?

 

DAN’S PUZZLE

x y = y x

For a given real x greater than one, the equation almost
always has two solutions for y. For example, for x = 2,
then y = 2 and y = 4 are both solutions. For x = 5.3, then
y
= 5.3 and y = 1.71586398526979... are both solutions.
But there is a real x greater than one for which there is
only one solution for y.

 

GREG'S PUZZLE

Bisect a line segment using only a compass (no straight edge).

 

MARTIN'S PUZZLE

Disprove the assertion that is an integer.

 

 

CUBBY'S PUZZLE

Prove the assertion that is an integer.

 

 

PETTIFERM'S PUZZLE

It seems that 2p–1 – 1 is always divisible by p if p is prime.
For example, for p = 5, then 25–1 – 1 = 24 – 1 = 15, which
is divisible by 5. For p = 7, then 27–1 – 1 = 26 – 1 = 63,
which is divisible by 7. Prove that 2p–1
– 1 is, in fact, divisible
by p if p is prime. Find a p that is not prime for which 2p–1
– 1
is divisible by p.