Puzzles

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and we'll e-mail you back the solution.

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

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?

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.)

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 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?

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

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

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

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

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)?

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.

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 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.)

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

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?

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.

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, 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.)

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

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.

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