You are to roll a die over and over until you get a six. What is the
expected total of all throws before you throw a six? By the "total" I
mean the sum of the die faces, for example the sequence 1-3-5-6 would
have a total of 9.
(Answer),
(Solution).
A cubic piece of cheese has been subdivided into 27 subcubes (so that it looks
like a Rubik's Cube). A mouse starts to eat a corner subcube. After eating
any given subcube it goes on to another adjacent subcube. Is it possible for
the mouse to eat all 27 subcubes and finish with the center cube?
(Answer),
(Solution).
What is the area of an equalateral pentagon of side 1? Answer many not be expressed in trigonometric functions. Hint: solve problem problem 47 first.
(Answer),
(Solution).
There exists a small town of 101 people in which
John and Richard are running for mayor. Each person in the town
(including the two running) cast their vote based on
the toss of a fair coin. As the ballots are counted
the results are reported vote by vote. The final result is
51 votes for Richard and 50 for John. What is the probability
that as the votes were being counted Richard was always ahead?
(Answer),
(Solution).
On average how often does the minute hand pass the hour hand on
an ordinary clock?
(Answer),
(Solution).
On 'The Price is Right' there is a game in which each contestant
makes a guess as to the value of a particular item. The
contestant who comes closest, without going over, wins. If all
contestants overbid then they start over until one of them
does win. Lets assume there are only three contestants (call them
x, y, and z) and that x goes first, then y, and then z. Each
player can hear the bids of all previous players. Also
assume that it is common knowledge that the value of the item
is a random variable with a uniform distribution over the range
of $1000 to $2000 (unless everyone overbids in which case the
range would be narrowed). Also assume that if everyone overbids
then all players will bid the same amount as before but relative to the new
interval. Also assume that bidding is done to the nearest penny. What should the
strategy of each player be given that x
assumes y and z are mathematicians and y assumes z is a mathematician.
(Answer).
It is your task to determine how high you can drop a billiard
ball without it breaking. There is a 100 story building and
you must determine which is the highest floor you can drop a
ball from without it breaking. You have only two billiard
balls to use as test objects, if both of them break before
you determine the answer then you have failed at your task.
How can you determine the breaking point in which the maximum
necessary dropping is at a minimum. Click on 'answer' for
just the minimum maximum and 'solution' for how to determine
the breaking point under this minimum.
(Answer).
(Solution).
What is the expected number of flips of a fair coin until you get two heads in a row?
What is the expected number until you get a head followed by a tail? Note: The
answers are not the same.
(Answer).
(Solution).
Create two six-sided dice, such that the probability of each sum from 2 to 12 is the same as two standard dice. Each side must have at least one dot. Negative numbers are not allowed. There is another answer besides two standard {1,2,3,4,5,6} dice.
(Answer).
(Solution).
What is the most pieces you form with n cuts of a pizza?
(Answer).
(Solution).
A man is standing on a rock in the middle of a circular lake of radius 1. There is a
tiger on the shore of the lake that can run four times as fast you can swim, however
the tiger can not swim. The tiger is hungry and always attempts to keep the distance
between the two of you at a minimum. How can you safely swim to shore?
(Answer).
The figure above is a square composed of 21 smaller squares. Each of the 21
smaller squares has a side of integer length and all 21 are different sizes.
Find any solution for the size of the 21 smaller squares.
(Answer)
(Solution)
Thanks to Terry W. Ryder of San Leandro, CA and
New Scientist magazine for the problem and Terry for the graphic.
A man has $1,000,000 he wishes to divide up in his will. He wants to give each person
named in his will an amount of money, in dollars, which is a power of 7 ($70=$1,
$71=$7, $72=$49, $73=$343, ...). He does not want to give more
than six people the same amount. How can he divide the money?
(Answer)
(Solution)
Assume two countries, call them the United States and Mexico, both have dual product economies. The
table below shows how many days it takes for one person in each country to produce a unit of food or a unit of
clothing.
Country
Food
Clothing
United States
1
2
Mexico
3
4
A man wishes to sell a puppy for $11. A customer wants to buy it but only has foreign
currency. The exchange rate for the foreign currency is 11 round coins = $15,
11 square coins = $16, 11 triangular coins = $17. How many of each coinage should
the customer pay?
(Answer),
(Solution).
Jack and Jill each have marble collections. The number in Jack's collection in
a square number (1,4,9,16, etc). Jack says to Jill, "If you give me all your marbles
I'll still have a square number." Jill replies, "If you gave me the number in my
collection you would still left left with an even square." What is the least
number of marbles Jack has?
(Answer).
A small town in Alaska is approaching winter. Because the soil will soon freeze they need
to dig enough graves in the town cemetary now in anticipation of the number of
deaths until the ground thaws in spring. The town's population is 1000 and it is
assumed that each person has a 1% chance of dying during the winter. What is the
least number of graves should they did so that the probability of having enough is at least
90%? What about 95% and 99%?
(Answer),
(Solution).
Each letter in this letter represents a specific and unique integer.
What does each letter stand for? Note that the M can not stand for 0, because then only four digits would be required for the sum.
(Answer),
(Solution).
A college student send the following letter to his parents:
SEND
+MORE
-----
MONEY
A casino offers to refund half your net losses if you bet only
on red or black in roulette.
The roulette wheel is the standard American variety that has
both a zero and double zero and pays even money. You must bet
the same amount every time. How many times should you bet to
maximize your expected return?
(Answer),
(Solution).
Consider a infinite number of parellel lines, spaced one inch apart from
each other. If you dropped a one inch toothpick at random on this set of
lines what is the probability the toothpick would cross a line?
(Answer),
(Solution).