that you manage to find something to purchase that costs
less than a dollar. If you have sufficient change in coins to cover the
how much more change in coins will you end up with if you pay with a
bill than if you paid with the coins? Explain your answer without using
Most and the Least This
problem is based on a common programming algorithm.There
are four coins of differing weights. Using a balance,
simultaneously determine the heaviest and the lightest coins in four
small isolated town is concerned about its ability to evacuate
the population in case of an emergency. It has a single bumpy dirt road
through it that can only handle traffic traveling up to 30 miles per
the road is upgraded to handle traffic up to 60 mph, why won't the
of evacuation double?
ordinarily buy several pounds of Qibx at $10.00 per pound.
Unfortunately, your regular supplier is out and won't be getting any
tomorrow. You have an immediate need for one pound of Qibx, so you go
another supplier. A one pound package goes for $12.00. You can also buy
package for $33.00, or $11.00 per pound. Which package is more
those who have never watched the television quiz show Jeopardy:
In the final round of the show, each contestant can wager any amount up
person's current winnings. If the person answers the question
amount wagered is added to the previous winnings. Otherwise it is
there are two contestants in the final round and that the
one in second-place enters the round with winnings of $12,000. The most
this person could end up with is $24,000. The leading contestant would
assured of at least a tie by starting the round with $24,000 or more.
the leader has less than $24,000 she can not be assured of at
least a tie. Suppose that we eliminate the case where the leader
question incorrectly and her opponent answers correctly. How much would
leader need to assure at least a tie?
is another problem that draws its inspiration from computer
recently, the Noomians had a thriving culture. There are
currently no speakers of the Noomian language. This highly literate
behind an extensive literature, including a book written in Noomian
to translate from Noomian into English. There is also a book written in
language of Narus telling how to translate from Noomian to Narus.
Language Publishers would like to produce a book written in
English that tells how to translate from Noomian into English. They
located one of the few remaining speakers of Narus. Unfortunately, the
gentleman speaks no other language. With considerable effort the people
Language have managed to convey their desire to produce an English
of the Noomian to English book and the man has agreed to do it. How
does he go
about writing the book?
is the size of the largest collection of
numbers from 1 to 100 with the property that no number in the
exactly twice as large as any other number in the collection?
What is the size of the smallest double-free collection of numbers from
100 which is complete, where by complete is meant that adding another
the collection will cause some number in the collection to be twice as
some other number in the collection?
Least One Divisor Shortly
after I thought of the previous
problem I read about this one, whose statement and solution are closely
related. It provides a nice contrast between the product of a hacker
myself and true genius.
This problem seems monstrously difficult but it has a simple solution.
First an easy problem. Find 50 numbers from 1 to 100 such that none of
divides evenly into any of the others. There are many solutions to
there were not, the second part of the problem would be easy. There is,
however, one solution that is fairly obvious.
Now show that 50 numbers is the best that can be done. That is, show
any 51 numbers from 1 to 100 one of them must be divisible by one of
Show that all numbers can be expressed as O*2n,
Use the insight gained from the previous demonstration to devise a
finding the 50 smallest numbers up to 100 such that no number divides
a 4 by 4 checkerboard, suppose that you can reverse the
colors of any row or any column. How many patterns can you create by
different combinations of rows and columns. You might want start on a 2
checkerboard or a 2 by 1 checkerboard.
that squares belonging to the intersection of a selected row
and a selected column may be thought of as being reversed twice, once
column and one by the row, and so will end up with their original color.
a. Young Tom has become infatuated with Vera, a toll
the highway that Tom rides to work on. There are 3 tollbooths and Tom
learned that the toll collectors are randomly assigned to a tollbooth
to the constraint that nobody works at the same tollbooth two days in a
simplify the problem assume that both Tom and Vera work 7 days a week.
Assume also that Tom is unable to see who is working in any tollbooth
other than the one where he is at. What strategy should Tom employ to
maximize the chances of seeing the object of his affection? On average,
how often will he see Vera?
b. This part is harder on both Tom and the reader. One day
notices Vera wearing an engagement ring. He is heartbroken, his only
consolation being that Vera has never taken notice of him. He wishes to
Vera as little as possible. What strategy should he now employ? What
chances on any particular day that the sight of Vera's countenance will
its toll upon Tom?
the function f(x) = x2
+ x +
1/x for x > 0. For large values of x the x2
as the function goes to infinity. As x goes to zero the 1/x2
term dominates as the function again goes to infinity. It is
to suppose that for some intermediate value of x the function attains a
value. To properly find this minimum value requires the use of calculus
given the information that the minimum is attained for a single value
of x, a
small insight into the nature of this function allows you to easily
value of x where the function has its minimal value. What is
the value of
x? What is the minimum value of the function?
many people are on the bus? A
bus route has 14 stops.For
each pair of stops
is exactly one passenger who gets on at the
first stop and also gets off at the second.For
example, there is one passenger who gets on at stop 2 and also gets
off at stop 8.Find an
equation for the
number of passengers on the bus after it unloads and loads passengers
what stop does the bus have the
maximum number of passengers as it pulls away frmo
many people are on the bus at that
first square-free century Using
simple algebra, show that the 36th century (3500 to 3599) is the first
century not to have a year that is an exact square. Hint:
The following will prove useful: (x+1)2 - x2 = 2x+1