Puzzle Answers
These are just my solutions: no guarantees they are correct!
I've put blank spaces between answers so you don't see other answers
by accident. If you want to print these, use this page without spaces instead.
Strange TripAnswer:
At the south pole.
Or, 100 miles south of any point on the circle
around the north pole that is 100 miles in circumference.
Or, 100 miles south of any such circle whose circumference is an
integral fraction of 100 miles.
Three Cards
Answer:
2/3. 2 of the 3 blue sides have blue on the other side.
You might think the odds are 50/50 because you could have one of
two cards, but you actually get a side at random, not a card, so you
are more likely to have the blue/blue card than the red/blue card.
Here is another way to think of it: When you pick a card at random,
the chances of it being the same on both sides (either all red or all
blue) are 2 out of 3, and this doesn't change just because you see
one of the sides. If they did, they'd change no matter which color
you saw, and it wouldn't even matter if you looked. So when you do
see the blue side, the red/red card is eliminated, but the blue/blue
card still has a 2/3 probability.
Monty Hall Problem
Answer:
You should switch. Your chances are 2/3 if you do. The initial
chances of your first choice were 1/3, and opening another door
without the prize doesn't actually change that, so the remaining door
now has a 2/3 chance because the chances of all possibilities must sum
to 1.
Many people think the chances are 50/50 because there are two doors
left, but that is not correct. If you're not convinced 2/3 is
correct, consider a 100 door version: you pick 1 door and Monty opens
98 other doors avoiding the one with the prize. Now you have a 99%
chance if you switch.
Monty Hall Problem
Answer:
Yes, your chances are 5/8 if you switch. The initial chances of your
first choice were 1/4, and opening another door without the prize
doesn't change that, so the remaining 2 doors now have a 3/8 chance
because the chances of all possibilities must sum to 1. But then when
Monty opens door 3, using the same reasoning, door 2 stays at 3/8
chance and so the remaining door changes to 5/8.
Two Envelopes
Answer:
This is a strange paradox because it appears that the expected value
is the average of double and half which is 1.25 X so you should
switch. However, this would be true for any value of X you find, so
it wouldn't even matter if you open the envelope. This same logic
suggests that you should also immediately switch back to the original
envelope again, which doesn't make sense at all.
The fault in this reasoning is that you are not actually given enough
information to know that double and half are equally probable. You
would need to know more about the process by which the amounts in the
envelopes were determined or what the probability distribution of
possible amounts is. For example, if there was a known limit for the
amounts, you would switch if and only if 2X was within that limit.
Note that it is impossible to have an even distribution of all
possible values without also having an upper bound. If the amounts
can go to infinity, there would have to be some uneven probability
distribution, and you would need to know what that was to solve the
problem properly.
Given the lack of any information like this, I believe the correct
answer is: It does not matter if you switch.
What's Next?
Answer:
Look at just the right-hand side of each figure.
Helium
Answer:
It moves backwards. The deceleration pushes the air in the car
forwards which causes the balloon to move in the other direction.
Urns with Balls
Answer:
74/99. Put 1 white ball in one urn and the rest in the other.
Bean Bag
Answer:
2/3. You might think it should remain at 50/50 since you didn't
change the initial state of the bag, but this is incorrect. There are
3 equally probable ways you could pick a white bean: 1st bean was
black and 2nd was picked, 1st bean was white and 2nd was picked, or
1st bean was white and also picked. 2 of those 3 have a white bean
left in the bag.
Four Trees
Answer:
Yes. If one or more trees are on a high hill they could form a
tetrahedron and all be equidistant.
Suicidal Spots
Answer:
They are all dead. Solve recursively: If only one had red he knows it
must be him. After he goes, the others know he must have seen only black
spots, so then they all go. If two have red, they see only one other red,
and when the one other doesn't go the first morning, they both go the
second morning, etc.
One Question
Answer:
"Which path would your brother tell me leads to the cannibals?" then
go that way. Or to ask just 1 person: "If I were to ask you which
path leads to the cannibals, what would you say?" and go the other way.
Random Hats
Answer:
If the 2 hats you see are the same, guess the opposite color,
otherwise pass. If all 3 players use this rule, it works 75% of the
time. It fails only when all the hats are the same color.
Northwest Spiral
Answer:
sqrt(2). The second pilot is always heading 45 degrees off from north, so he
always needs to travel sqrt(2) farther to make the same northerly
progress.
Odd Ball
Answer:
First weigh 4 balls vs 4.
If they balance:
You know the odd ball is one of the other 4 and you can divide-by-2
search from there by just weighing 2 of them and then 1 against the
normal balls.
If the initial 4 vs 4 don't balance:
Name those 8 balls H or L for heavier and lighter as appropriate,
and call the other normal balls N.
Balance (2H + 2L) vs (1L + 1H + 2N).
If that balances:
You know the odd ball is one of the other two H or L, which you can
determine with a final weighing as above.
If that doesn't balance:
You have narrowed it down to 3 possible balls, because the tipping
direction must be consistent with the previous weighing.
If the (2H + 2L) side is heavier:
It must be one of those 2H or else the 1L from the other side.
Weigh those 2 H balls against each other:
If it balances: the 1L from the other side is the odd ball.
If it tips: the heavier one is the odd ball.
If the (2H + 2L) is lighter, do the symmetrical thing as above reversing L and H.
There are variations on this scheme that also work.
Bags of Gold
Answer:
Take 1 piece from the 1st bag, 2 pieces from the 2nd bag, and so on up to
9 pieces from the 9th bag. The total weight W would normally be 45. If
it is, the 10th bag has the lighter pieces. Otherwise the number of
the guilty bag is (45 - W) x 10.
Fair Cake
Answer:
A cuts off a piece. B has the option to reduce that piece and take
it, and then C has the same option. The remainder of the cake is
split between the other 2 people with the usual "one cuts and the
other chooses".
Alternate recursive method: Starting with approximate thirds, each
person can swap with either of the other 2 but then must give at least
a crumb back to whoever he swaps with. This cycles until nobody wants
to swap anymore.
Light Switches
Answer:
Start with all switches off. Turn switch A on and wait 10 minutes or
so. Turn A off and B on. Go into the room. Light A is off and warm,
B is on, and C is off and cold.
Grid of Tiles
Answer:
No. If the grid is checkered, each tile must cover 1 black and 1 red
square. However both missing corners are the same color, so there
must be 2 squares with the other color that can never be covered.
1x3 Tiles
Answer:
Yes. The 1x1 tile can only be in a 3,3 position. If you label the
squares of the grid as follows:
X . , X . , X .
, X . , X . , X
. , X . , X . ,
X . , X . , X .
, X . , X . , X
. , X . , X . .
X . , X . , X .
, X . , X . , X
Each 1x3 tile must cover exactly one of each type. There is one more
X than dots or commas so the 1x1 square must cover some X position. If
you turn this pattern the other way, the same should still be true, and
there are only four X positions that remain X after you turn it:
, X . , X . , X
X . , X . , X .
. , X . , X . .
, X . , X . , X
X . , X . , X .
. , X . , X . .
, X . , X . , X
X . , X . , X .
Here is an example solution:
- - - | | - - -
- - - | | - - -
| | X | | - - -
| | - - - - - -
| | - - - - - -
| | | | | - - -
| | | | | - - -
| | | | | - - -
Cheese CubesAnswer:
No. If the grid is checkered so adjacent cubes are different colors,
the mouse must always alternate colors. There are an odd number of
cubes so the mouse must end on the same color as he starts, but the
center is not the same color as the corners.
Measuring with Jugs
Answer:
To measure 4 liters: fill the large jug and pour off 3L by
filling the small jug, to leave 2L. Empty the small jug, and transfer
the 2L over to it. Then fill the large jug again and pour off 1L by
topping off the smaller jug. That leaves 4L in the large jug.
To measure 1 liter: fill the small jug and pour that 3L into the large
jug. Fill the small jug again and pour off 2L by topping off the
large jug. That leaves 1L in the small jug.
Burning Fuses
Answer:
Light both ends of the 10 minute fuse at once. When those burning
ends meet somewhere in the middle, light the 15 minute fuse. When
that is finished burning, 20 minutes are up.
Pocket Change
Answer:
16 cents: 1 penny, 1 nickel, and 1 dime.
Gloves and Germs
Answer:
Put both gloves on the same hand and squeeze the 1st culture. Take
the outer glove off and turn it inside out (its now dirty inside and
clean outside). Use the glove that is still on to squeeze the 2nd
culture. Put the second glove back on over the first (so their dirty
sides touch) and squeeze the 3rd culture.
Each squeeze makes 2 sides of a glove dirty (inside and out). 3
squeezes makes 6 sides dirty, but you only have 4 total sides.
Therefore you know you must somehow re-use the same side that touches
your skin for all 3 squeezes, as above.
Mixed Up Liquids
Answer:
They are equal. Call the initial amount of liquids in the jars J,
measured in teaspoons. When you add a teaspoon of milk to the water,
the milk ratio of the first jar becomes M1 = 1/(J+1) and the water
ratio is W1 = J/(J+1). Then, when you add a teaspoon of the mixture
back to the water, the resulting water ratio is W1/J = 1/(J+1) = M1.
Feynman's Sucking Sprinkler
Answer:
The net change in momentum is zero, so it should not spin.
The sucking might tend to make it go backwards, but then the
change in momentum at the corners would cancel that.
Train Full of Water
Answer:
(Subject to debate.) The train moves to the left as the water moves to
the right to get to the spout. But then the water leaving moves to the
left and pushes the train back towards the right to conserve momentum, so
the train ends up with a slight rightward velocity.
Bubbles in Space
Answer:
Towards each other.
Six Chop Sticks
Answer:
Make a 3d tetrahedron. Each of its 4 sides is a equilateral triangle.
Stick Boxes
Answer:
If 4 boxes are made from 16 sticks, you know that no boxes can
share edges, only corners.
Fish Sticks
Answer:
Ten Points
Answer:
Arrange the points in a star shape, with 5 at the star's points, and 5
at the inner intersections.
Connect the Dots
Answer:
Wire Cube
Answer:
4. Each cube vertex is formed by 3 edges so you need a minimum of one
wire-end for each. There are 8 vertices and 4 wires have 8 ends.
Bookworm
Answer:
1602 (or 1600 if you don't count the first and last pages). Books are
normally arranged on a bookshelf in order starting with volume 1 on
the left, and with their binding facing outwards. Viewing books this
way, the first page of a book is on its right and the last page is on
its left, so the worm actually only eats through the middle 8 volumes.
Five Hats
Answer:
Assume they are looking left in the order A,B,C. If C saw two black
hats he would know he had white. If B saw one black and C didn't say
anything, he would know he had white. But neither C nor B say
anything, so then A knows he must have white.
Pop Quiz
Answer:
This inductive reasoning leads you to claim the surprise can't be on any
day, and therefore doesn't actually differentiate any of them.
Darts
Answer:
2/3. This is just another way of saying: what are the chances
that your third dart is not the best.
Girl Babies
Answer:
The male to female ratio is 50/50. The odds of each new baby are
always 50/50, and a rule for when families stop having children does
not affect those odds.
The population size also stays the same, because there are 2 children
per family on the average. They all have 1 girl, and the boy/girl
ratio is 50/50 so they must also have an average of 1 boy.
Another way to calculate this is as follows: All families have 1 girl.
Prior to that, 1/2 of the families also have at least one boy, 1/4
have another boy, 1/8 have another, etc. The sum of the series (1/2 +
1/4 + 1/8 + 1/16...) equals 1. So on the average its just 1 girl and
1 boy.
Hotel Bellboy
Answer:
You wouldn't add the bellboys $2 to the $27, you would subtract it.
They paid $27, the bellboy kept $2, and $25 was for the room.
Doctor Who
Answer:
The doctor is his mother. (Shame on you if you didn't think of that.)
Non-Self Containing Sets
Answer:
If this superset (of all sets that don't contain themselves) contains
itself, then it shouldn't, and visa versa.
Manhole Covers
Answer:
So they don't fall in. A square cover nearly the same size and shape
as the hole, for example, could fit into the hole diagonally.
Mirrors
Answer:
Mirrors actually flip things front-back. Flipping left-right and then
rotating about a vertical axis gives the same transform. A "mirror
image" is more easily perceived this way, probably because humans are
left-right symmetrical. You could also think of this transform as
flipping up-down and then rotating about a horizontal axis but this is
harder to visualize unless you turn your face sideways first.
Centrifuge
Answer:
Yes. You can independently balance 3, and then 2. Put 3 in equally
spaced from each other, such as in slots 1, 5, and 9. Then put in the
other 2 opposite each other, such as in slots 2 and 8.
1000 Doors
Answer:
The doors whose number equals a perfect square are open. Those with
an odd number of factors, including 1 and the number, are open, and
only perfect squares have an odd number of factors.
Square of Bugs
Answer:
1 meter. A bug's movement remains perpendicular to the next bug's, so
the total distance is the same as if the bug being chased was
stationary. They start out 1 meter apart so that is the total
distance traveled when they meet in the center.
Another solution method: They must meet in the center, and the
distance from a corner to the center is 1/sqrt(2). The bugs always
head at 45 degrees relative to the center, so they will travel
a total of sqrt(2) times that distance.
A Bee and Two Trains
Answer:
50 miles. The trains travel for 2 hours until they meet, and the bee
flies at 25 mph the whole time, so he flies 50 miles. (Hope you didn't
try to calculate each zig and zag!)
Backwards Bee and Two Trains
Answer:
There is no way to know. Trick question.
Water Levels
Answer:
The stone sinks and then displaces less water than when it was being
floated by the boat, so the water level falls.
The log floats and continues to displace the same amount of water as
its weight, so the water level stays the same.
Pieces of Stone
Answer:
1, 3, 9, and 27. You can put multiple pieces on the scale to add
their weights, or put pieces on the other side to subtract them.
Breaking Balls
Answer:
You can do it in 14 drops. You can't use a normal divide-by-2 search
here because once the first ball breaks you'll need to search all the
untested floors below from bottom to top. To find the best strategy,
work backwards from fewer floors:
1 drop allows you to test 1 floor.
2 drops can test 3 floors: Test the 2nd floor first. It it breaks,
test the bottom floor with the other ball. If not, test the top floor
with either ball.
3 drops can test 6 floors: Test the 3rd floor first. If it breaks
test the bottom 2 floors in order with the other ball. If not, test
the top 3 floors as described above.
4 drops can test 10 floors: Test the 4th floor first. If it breaks
test the bottom 3 floors in order with the other ball. If not, test
the top 6 floors as described above.
N drops allows you to test the number of floors equal to the sum of 1
to N. 14 drops can test 105 floors: Test the 14th floor first. If it
breaks test the bottom 13 floors in order with the other ball. If
not, test the 27th floor next (14+13), and so on.
For any given drop, the number of drops you'll need after that should
be the same if that ball breaks or not (or sometimes they are off by 1
if your total is not an even sum-of-1-to-N).