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Math 1350 Project Problems for Chapters 1, 2, and 9(Due by EOC Oct. 23)
Caution, This Induction May Induce Vomiting!
1. a) Observe that 2 3 4
1 2 2 33
,
3 4 51 2 2 3 3 4
3
, and
4 5 6
1 2 2 3 3 4 4 53
.
Use inductive reasoning to make a conjecture about the value of
1 2 2 3 3 4 1n n .
Use your conjecture to determine the value of 1 2 2 3 3 4 100,000 100,001 .
b) Observe that 2
3 12
1 3 , 2
5 12
1 3 5 , and 2
7 12
1 3 5 7 .
Use inductive reasoning to make a conjecture about the value of 1 3 5 2 1n .
Use your conjecture to determine the value of 1 3 5 1,999,999 .
c) Observe that 1 1
1 2 1 1
,
1 1 2
1 2 2 3 2 1
, and
1 1 1 3
1 2 2 3 3 4 3 1
.
Use inductive reasoning to make a conjecture about the value of
1 1 1 1
1 2 2 3 3 4 1n n
.
Use your conjecture to determine the value of 1 1 1 1
1 2 2 3 3 4 999 1,000
.
An Even Sum.
2. Find the one’s digit of the following sum: 20109 81 729 9 .
{Hint: Start small, and look for a pattern.}
Facebook Shmacebook.
3. After graduation exercises, each senior gave a snapshot of himself or herself to every other
senior and received a snapshot in return. If 2,000,810 snapshots were exchanged, how many
seniors were in the graduation class?
Hint: Start with small classes and look for a pattern.
The Last Two Standing
4. What are the final two digits of 20177 ?
{Hint: Look for a pattern:
Power of 7 Final two digits
17 07 07
27 49 49
37 343 43
47 2401 01
57 16807 07
67 117649 49
.}
The Last One Standing.
5. Find the ones digit of 2421 378313 17 .
{Hint: Look for a pattern:
Powers of 13 One’s-digit Powers of 17 One’s digit
113 3 117 7
213 9 217 9
313 7 317 3
413 1 417 1
513 3 517 7
613 9 617 9
.}
Don’t Give Up; Don’t Ever Give Up!
6. Given that 11 11f and
13
1
f xf x
f x
for all x , find 2018f . First find
14 , 17 , 20 ,f f f .
{Hint: Look for a pattern:
n f(n)
11 8 3 1 11
14 8 3 2 5
6
17 8 3 3 1
11
20 8 3 4 6
5
23 8 3 5 11
.}
I Hope That You Are A Digital Computer?
7. What is the ones digit of 1! 2! 3! 999! ?
{Hint: See what happens to the one’s digits:
Factorial One’s-digit One’s digit of the sum
1! 1 1
2! 2 3
3! 6 9
4! 4 3
.}
Dots And Dashes, But It’s Not Morse Code.
8. The diagram shows a sequence of shapes 1 2 3 4, , , ,T T T T . Each shape consists of a number of
squares. A dot is placed at each point where there is a corner of one or more squares.
1T 2T 3T 4T
Shape number of rows, n number of squares, S number of dots, D 2D n
1T 1 1 4 3
2T 2 4 10 6
3T 3 9 18 9
4T 4 16 28 12
a) Use inductive reasoning to find a formula for S in terms of n.
b) How many squares are in shape 25T ?
c) Use inductive reasoning to find a formula for D in terms of n.
{Hint: Notice that 2D n in the last column is always a multiple of 3.}
d) How many dots are in shape 25T ?
A Sequence Perfect For The Lucasion.
9. The 4th term of a sequence is 4, and the 6th term is 6. Every term after the 2nd one is the sum of
the two preceding terms.
a) Find the 8th term of this sequence.
b) Find the 1st term of this sequence.
Oh, Don’t Be a Square, Or A Triangle For That Matter.
Let’s compare the sequence of values given by the formula a n n n
, where
means rounded to the nearest whole number, with the sequence of non-square numbers.
1 1 1 2 2a , 2 2 2 3.414... 3a
,
3 3 3 4.732... 5a , 4 4 4 6 6a
,
5 5 5 7.236... 7a , 6 6 6 8.449... 8a
,
7 7 7 9.645... 10a . Assuming that this continues, the 100th non-square
number is 100 100 110 110 .
a) Find a formula for the non-triangular numbers of the form a n n An
for some whole
number A.
b) Use your formula in part a) to predict the 100th and 200th non-triangular numbers.
10.
Geoarithmetric Sequences.
11. Solve the following arithmetic/geometric sequence problems.
a) In a geometric sequence of real number terms, the first term is 3 and the fourth term is
24. Find the common ratio.
b) Find the seventh term of a geometric sequence whose third term is 94 and whose fifth
term is 8164 .
c) For what value of k will 4, 1,2 2k k k form a geometric sequence?
d) Find the second term of an arithmetic sequence whose first term is 2 and whose first,
third, and seventh terms form a geometric sequence.
e) Is there a geometric sequence containing the terms 27, 8, and 12 in any order and not
necessarily consecutive? If so, give an example. If not, show why.
f) Is there a geometric sequence containing the terms 1, 2, and 5 in any order and not
necessarily consecutive? If so, give an example. If not, show why.
g) Find all numbers a and b so that 10, , ,a b ab are the first 4 terms of an arithmetic sequence.
Now What Was That Last Digit?
12. Peggy is writing the numbers 1 to 9,999.
a) If she stops to rest after she has written a total of 150 digits. What is the last digit that she
wrote?
b) If she stops to rest after she has written a total of 1,000 digits. What is the last digit that
she wrote?
{Hint:
Quantity Total number of digits
1 digit numbers(1-9) 9 9
2 digit numbers(10-99) 90 180
3 digit numbers(100-999) 900 2,700
4 digit numbers(1,000-9,999) 9000 36,000
.}
Book, Line, and Thinker.
13. Pages of a book are numbered sequentially starting with 1.
a) If the total number of digits used in the numbering of the pages is 159, then how many
pages are in the book?
b) If the total number of digits used in the numbering of the pages is 1578, then how many
pages are in the book?
c) If the total number of digits used in the numbering of the pages is 5893, then how many
pages are in the book?
{Hint:
Pages Total number of digits
1-9 9
10-99 180
100-999 2,700
1,000-9,999 36,000
.}
Middleaged At 40?
14. The counting numbers are written in the pattern below. Find the middle number of the 40th
row.
1
2 3 4
5 6 7 8 9
10 11 12 13 14 15 16
Odds, Evens, What’s The Difference?
15. a) What do you get if the sum of the first 8,000,000,000 positive odd integers is subtracted
from the sum of the first 8,000,000,000 positive even integers?
{Hint: 2 4 6 8 16,000,000,000 1 3 5 7 15,999,999,999 }
b) What do you get if you subtract 8,000,000,000 from the sum of the first 8,000,000,000
even numbers?
Pyramid Power.
16. In the following pyramid, the number in each stone is found by adding together the numbers
in the two stones below it. Complete the pyramid.
It’s Magic!
17. Complete the magic square, containing the digits 1-9, so that each column, row, and diagonal
adds up to 15.
2 6
9
Triangulate Your Answer.
18. What number is missing from the third triangle? Why?
{Hint: How does 7 6 compare to 5 9 ?}
2 8 6
7 14
6
5
1 7 9
9
8
8 8
8
7
1 6 10
It’s More Magical!
19. The numbers in each row, column, and diagonal add up to 34. All the numbers from 1 to 16
appear, but only once each. Fill in the missing numbers.
16 6
4 10
14 8
2 12
Insert Two Digits, But Not In Your Nose.
20. Write a single digit in each box to make the product of the 3-digit number and the 2-digit
number calculation correct.
34 x 3 = 26,424
The method of finite differences can be used to produce formulas for lists of numbers. For
example, if the list of numbers is 5,7,9,11,13,…, then the list of first finite differences is the list
of differences of consecutive numbers: 7 5 9 7 11 9 13 11
2 , 2 , 2 , 2 ,
or more simply, 2,2,2,2,…. Whenever
the list of first finite differences is a repetition of the same number, it means that the original list
of numbers can be produced by a formula of the form an b , where n represents the position in
the list. Here’s why:
original a b 2a b 3a b 4a b
first finite differences a a a
In fact, the repeated number in the first finite differences will always be the coefficient of n.
In our example, it must be that 2a and 5a b . So we get that 2a and 3b , and a formula
for the list of numbers 5,7,9,11,13,… is 2 3.n Sometimes the list of first differences is not a
repetition of the same number. An example of this is the list 3,9,19,33,51,73,…. The list of first
finite differences is 6,10,14,18,22,…. As you can see it’s not a repetition of the same number.
Now let’s calculate the list of second finite differences: 4,4,4,4,…. Whenever the list of second
finite differences is a repetition of the same number it means that the original list of numbers can
be produced by a formula of the form 2an bn c , where again n represents the position in the
list. Here’s why:
original a b c 4 2a b c 9 3a b c 16 4a b c 25 5a b c
first finite differences 3a b 5a b 7a b 9a b
second finite differences 2a 2a 2a
In fact, the repeated number in the second finite differences will always be twice the coefficient
of 2n . In our example, it must be that 2a , 3a b c , and 4 2 9a b c . Substituting the
value of a into the next two equations leads to the system 1
2 1
b c
b c
. Subtracting the first equation
from the second equation leads to 0b , so the values are 2a , 0b , and 1c . So a formula
for the list of numbers 3,9,19,33,51,73,… is 22 1n . Similar results hold for higher order
differences which are repetitions of the same number. You may use the method of finite
differences to solve the following four problems.
What’s the Diff?
21. Find a formula for each of the following lists of numbers:
a) 2,5,8,11,…
b) 3 5 72 2 2,2, ,3, ,
c) 4,7,12,19,28,
d) 3,7,13,21,31,43
e) 1, 1 2 , 1 2 3 , 1 2 3 4 , 1 2 3 4 5 ,
f) 1, 1 3 , 1 3 5 , 1 3 5 7 , 1 3 5 7 9 ,
g) 1,15,53,127,249,…
h) 1, 1 4 , 1 4 9 , 1 4 9 16 , 1 4 9 16 25 , 1 4 9 16 25 36 ,
You Want Me To Cut The Pizza Into How Many Slices?
22. A chord is a line segment joining two points on a circle. Here, n is the number of chords.
Associated with each number of chords, n, is the maximum number of regions that the circle is
divided into by the n chords. For the first five numbers of chords, the list of the maximum number
of regions is 2,4,7,11,16 . Use the method of finite differences to find a formula for the maximum
number of regions that a circle can be divided into using n chords, and use the formula to predict
the maximum number of regions a circle can be divided into using 60 chords.
1n 2n 3n 4n
5n
A Checkered Present.
23. a)How many squares can you find on a 8 8 checkerboard?
b) an n n checkerboard?
Hint: Start with smaller boards and look for a pattern.
1 large, 4 medium, and 9 small
1 + 4 + 9 = 14
1 large and 4 small
1 + 4 = 5
1 extra large, 4 large, 9 medium, and 16 small
1 + 4 + 9 + 16 = 30
Have You Done Your Christmas Shopping Yet?
24. In the song “The Twelve Days of Christmas”, how many gifts in all did “my true love give to
me”?
{On the first day: 1 gift, On the second day: 3 gifts, On the third day: 6 gifts. For the first
three days it’s a total of 10 gifts. Find the total for all twelve days.}
Oh Brother! No, Oh Sister!
25. In a family, each boy has twice as many sisters as brothers, and each girl has two more sisters
than brothers. How many brothers and sisters are in the family?
{Hint: Let b be the number of brothers and s the number of sisters. Now write down some
equations.}
Exactly How Do You Want Your Million?
26. Find a positive number that you can add to 1,000,000 that will give you a larger value than if
you multiplied this number by 1,000,000? Find all such numbers.
{Hint: Let the positive number be x, and solve 1,000,000 1,000,000x x .}
Interesting Is In The Eye Of The Beholder
27. There is an interesting five-digit number. With a 1 after it, it is three times as large as with a
1 before it. What is the number?
{Hint: If x is the five-digit number, then , 1 10 1,1 100,000x abcde abcde x abcde x .}
Total Gifts 1 4 10 20 35 56
1st difference 3 6 10 15 21
2nd difference 3 4 5 6
3rd difference 1 1 1
Twenty-one, But Not Blackjack.
28. Find the 21-digit number so that when you write the digit 1 in front and behind, the new
number is 99 times the original number.
{Hint: If x is the 21-digit number then x abcdefghijklmnopqrstu ,
then 1 10 1abcdefghijklmnopqrstu x
and 1 1 10,000,000,000,000,000,000,000 10 1abcdefghijklmnopqrstuv x .}
Solving Without Completely Solving.
29. If 2a b and 2 2 3a b , then find 8 8a b .
{Hint: Start squaring and substituting.}
Gerry Benzel’s Favorite Problems.
30. a) A bottle and a cork together cost $1.10. If the bottle costs $1.00 more than the cork, what
does the cork cost?
{Hint: Let x be the cost of the cork and y the cost of the bottle.}
b) A shirt and a tie sell for $9.50. The shirt costs $5.50 more than the tie. What is the cost
of the tie?
It Squares; It Cubes; It Does It all!
31. If 3 3 10a b and 5a b , then what is the value of 2 2a ab b ?
{Hint: 2 2 3 3a b a ab b a b .}
Getting Solutions Without Actually Solving.
32. Notice that
2x a x b x a b x ab
3 2x a x b x c x a b c x ab ac bc x abc
4 3 2x a x b x c x d x a b c d x ab ac ad bc bd cd x
abc abd acd bcd x abcd .
a) Use inductive reasoning to determine the value of the coefficient of 1nx and the constant
term in the expansion of the following product: 1 2 nx a x a x a .
b) Use the previous result to determine the sum of the seventeenth powers of the 17 solutions
of the equation 17 3 1 0x x .
{Hint: The Fundamental Theorem of Algebra guarantees that the equation 17 3 1 0x x
has seventeen solutions(counting duplicates). The seventeen solutions of
17
1 2 17 3 1 0x a x a x a x x are 1 2 17, , ,a a a . So adding the
seventeen equations together yields:
17
1 1
17
2 2
17
17 17
17 17 17
1 2 17 1 2 17
3 1 0
3 1 0
3 1 0
3 17 0
a a
a a
a a
a a a a a a
. Now use the previous result.}
Two Squares Don’t Get Along – A Difference Of Squares!
33. If 20,000 20,000 5x y , 10,000 10,000 4x y , 5,000 5,000 3x y , 2,500 2,500 2x y , and 2,500 2,500 1x y , then find the value of 40,000 40,000x y .
Hint: 2 2a b a b a b .
Real-valued Function, What’s Your Function?
34. a) Suppose that f is a real-valued function with 20182 3x
f x f x for all 0x . Find
2f . {Hint: Plug in 2 and 1009, and solve for 2f .}
b) Suppose that f is a real-valued function with 21
23
x
f xf x
x for all 0x . Find
2f .
I Cannot Tell A Fib(onacci), My Name Is Lucas.
35. If 1 1f , 2 3f , and 1 2f n f n f n for 3,4,5,n
a) What is the value of 12f ?
{Hint: 3 2 1 1 3 4f f f , 4 3 2 4 3 7f f f ,…Keep going.}
Amazingly, f can be represented as n nf n x y for 1,2,3,4,5,n
b) Find the values of x and y .
{Hint: 1 , 1 1f x y f , 2 22 , 2 3f x y f , this should be enough to find
values of x and y.}
c) Alicia always climbs steps 1, 2, or 4 at a time. For example, she climbs 4 steps by
1-1-1-1, 1-1-2, 1-2-1,2-1-1,2-2, or 4. In how many ways can she climb 10 steps?
{Hint: If f n represents the number of ways to climb to the nth step, then
1 2 4f n f n f n f n .}
He Said, She Said.
36. If each different letter represents a different digit, find the two-digit number HE so that
2
HE SHE .
{Hint: 2 2
100HE SHE HE HE S , so 2
100 0HE HE S , and the quadratic
formula leads to 1 1 400
2
SHE
. Try this out with 1,2,3,4,5,6,7,8,9S .}
Highs And Lows In The Classroom.
37. a) The grades on six tests all range from 0 to 100 inclusive. If the average for the six tests is
93, what is the lowest possible grade on any one of the tests?
{Hint: 1 2 3 4 5 61 2 3 4 5 693 558
6
T T T T T TT T T T T T
1 2 3 4 5 6558T T T T T T , so 1T will be as small as possible when
2 3 4 5 6T T T T T is as large as possible.}
b) If the average for the six tests is 16, what is the highest possible grade on any one of the
tests?
Even So, It’s Odd.
38. Show that for every positive integer n, 2 3 8n n is even.
{Hint: n is either even or odd, so take it from here.}
It All Adds Up To Something.
39. a) There are 120 five-digit numbers that use all the digits 1 through 5 exactly once. What is
the sum of the 120 numbers?
Hint:
12345
12354
54321
120 numbers
How many of each digit occur in each column?
b) If the digits can be repeated, then there are 3,125 five-digit numbers that can be formed.
What is the sum of the 3,125 numbers?
It’s All In The Ones.
40. Find all the integer solutions of the equation 2 5 27x y , or show that there aren’t any.
{Hint: What are the possible ones digits of 2x ? What are the possible ones digits of 5y ?}
Just Your Average Airplane Flight.
41. An airplane travels between two cities. The first half of the distance between the two cities
is travelled at an average speed of 600 mph. The second half of the distance is travelled at an
average speed of 900 mph. Find the average speed of the airplane for the entire trip.
{Hint: distance travelled
average speedtime of travel
.}
Shifty Four.
42. The leftmost digit of a 6 digit number is 4. If it’s shifted to be the rightmost digit, the new
number is one-fourth of the original number. What’s the original number?
{If 4x abcde , then 4 10 400,000 4abcde x .}
Shiftier Six.
43. The leftmost digit of a number is 6. If it’s shifted to be the rightmost digit, the new number
is one-fourth of the original number. What’s the original number?
{Try two-digit numbers, three-digit numbers,…}
You’re A Real Square, Man.
44. A man born in the year 2x died, on his 89th birthday, in the year 2
1x . In what year was
he born?
Who Needs Logarithms?
45. For all positive numbers a and b, a function f satisfies f ab f a f b . If 2f x
and 5f y , then find the value of 100f in terms of x and y.
{Hint: 2 2100 2 5 .}
Zero Is My Hero.
46. If a and b are two unequal numbers, and ax bx , then find the exact numerical value of
3x
a b .
It’s As Easy As 123… .
47. If the digit 9 is written to the right of a certain number, that number is increased by
111,111,111. Find the number.
{Hint: If the original number is x , then 10 9x is the new number.}
How Long John?
48. John was three times as old as his sister 2 years ago, and five times as old 2 years before that.
In how many years will John be twice as old as his sister?
With Friends Like You… .
49. A group of friends decide to pool their money to buy a wedding gift. They find that if each
of them pays $8, they will have $3 more than the price of the gift, and if they each pay $7,
they will have $4 less than the price of the gift. What’s the size of the group of friends?
What’s the price of the gift?
Sometimes You Got To Kiss A Lot Of Frogs.
50. There are some princes and frogs in a fairy-tale. As a group, they have 35 heads and 94 feet.
How many princes, and how many frogs are there?
Find X + Y + Z PDQ.
51. If 2x y z , 1y z x , and 4z x y , then find the value of x y z .
{Hint: Add the three equations together.}
Don’t Shoot The Messenger.
52. a) Two armies are advancing toward each other, each at 1 mph. When the two armies are
10 miles apart, a messenger leaves the first army and races toward the second army at 9
mph. Upon reaching the second army, the messenger immediately turns around and races
back to the first army at 9 mph. How many miles apart are the two armies when the
messenger returns to the first army?
{Hint: The distance between the messenger and the second army is decreasing at the rate
of 10 mph, since they are originally 10 miles apart, it takes the messenger 1 hour
reach the second army. The distance between the two armies is decreasing at the
rate of 2 mph, so while the messenger is going from the first army to the second
army, the distance between the two armies decreases to 8 miles.}
b) Solve the same problem, except this time the messenger travels at 4 miles per hour.
Finally, A Light A The End Of The Tunnel.
53. A train which is 1 mile long is traveling at a steady speed of 20 miles per hour. It enters a
tunnel 1 mile long at noon. At what time does the rear of the train emerge from the tunnel?
(Both ends of the tunnel are in the same time zone!)
{Hint: How far does the front of the train have to travel?}
A European Sampler.
54. A box contains 8 French books, 12 Spanish books, 9 German books, 15 Russian books, 18
Italian books, and 10 Chinese books. What is the fewest number of books you can select
from the box without looking to be guaranteed of selecting at least 10 books of the same
language?
{Hint: What is the largest number of books you can select and still not have 10 books of the
same language? The answer to the problem is 1 more than the answer to the previous
question.}
Sorry, I Can’t Give You Change For A Dollar.
55. a) What is the largest amount of money in U.S. coins(pennies, nickels, dimes, quarters, but
no half-dollars or dollars) you can have and still not have change for a dollar?
{Hint: It’s more than 99 cents. For instance: 3 quarters and 3 dimes is $1.05, but you
can’t make change for a dollar.}
b) A collection of coins is made up of an equal number of pennies, nickels, dimes, and
quarters. What is the largest possible value of the collection which is less than $2?
c) Trina has two dozen coins, all dimes and nickels, worth between $1.72 and $2.11. What
is the least number of dimes she could have?
Officer, I Got The License Plate Number, But I Was Lying On My Back.
56. The number on a license plate consists of five digits. When the license plate is turned upside-
down, you can still read a number, but the upside-down number is 78,633 greater than the
original license number. What is the original license number?
{Hint: The digits that make sense when viewed upside-down are 0, 1, 6, 8, and 9.
1st digit 2nd digit 3rd digit 4th digit 5th digit
Upside-down plate
Original plate
Difference of the plates 7 8 6 3 3
The first digit of the original plate must be the flip of the fifth digit of the upside-down
plate. The second digit of the original plate must be the flip of the fourth digit of the
upside-down plate. The third digit of the original plate must be the flip of the third
digit of the upside-down plate. For the one’s digit in the difference, 9-6, 3 or 11-8}
Life Is Like A Box Of Chocolate Covered Cherries.
57. a) Assume that chocolate covered cherries come in boxes of 5, 7, and 10. What is the largest
number of chocolate covered cherries that cannot be ordered exactly?
{Hint: If you can get five consecutive amounts of cherries, then you can get all amounts
larger. Here’s why: Suppose you can get the amounts 27,28,29,30,31, then by the
addition of the box of size 5, you can also get 32,33,34,35,36 , and another addition of
the box of size 5 produces 37,38,39,40,41and so on. This would also be true of seven
consecutive amounts and ten consecutive amounts, but five consecutive amounts would
occur first. So look for amounts smaller than the first five consecutive amounts.}
b) Do the same problem, except the cherries come in boxes of 6, 9, and 20.
Let’s Not Go Overboard.
58. This problem in one form or another has been around for nearly 2,000 years – if not longer.
Josephus, a Jewish historian of the first century A. D., mentions it. In all its forms the problem
seems to have a violent streak. Here’s our version of it: One-hundred sailors are arranged
around the edge of their ship. They hold in order the numbers from 1 to 100. Starting the
count with number 1, every other sailor is pushed overboard into the cold North Atlantic
waters until there is only one left – the survivor.
a) What number do you want to be holding in order to be the survivor?
b) How many times will the survivor be skipped during this process?
c) Find the number of the last sailor to be pushed overboard.
d) Find the number of the second to last sailor to be pushed overboard.
{Hint: Consider the problem for smaller numbers of sailors and work up: For example
with 10 sailors you have
The survivor, 5, is skipped three times. The last sailor to be pushed overboard is
9, and the next to last sailor pushed overboard is 1.}
1 2
3
4
5 6
7
8
9
10
first
elimination
1
3
5 7
9
second
elimination
1
5
9
third
elimination
5
the
survivor
Put A Sock In It Brad And Angelina.
59. I) Mr. Smith left on a trip very early one morning. Not wishing to wake Mrs. Smith, Mr.
Smith packed in the dark. He had 6 pairs of socks that were alike except for color, and his
socks came in six different colors. Find the least number of socks he would have had to
pack to be guaranteed of getting
a) at least one matching pair of socks. b) at least two matching pairs of socks.
c) at least three matching pairs of socks. d) at least four matching pairs of socks.
{Hint: He could actually pack as many as 6 socks and still not have a matching pair.
1 1 1 1 1 1
Color 1 Color 2 Color 3 Color 4 Color 5 Color 6
II) If he had 12 pairs of socks(2 pair of each of the six different colors), answer the same 4
questions.
Stick puzzles involve rearranging, removing, or adding sticks in order to accomplish the
requirements of a problem. In the following problems, you might want to use the following
suggestions:
This arrangement of three sticks can be used to represent a square root:
A stick representing an over bar can be used to multiply a Roman numeral by 1,000:
represents 5,000
Just Stick It.
60. a) Move one stick to make a true equation.
b) Remove two sticks to make a true equation.
c) Move one stick to make a square.
d) Move one stick to make a true equation.
e) Move one stick to make a true equation.
Stick It, Again.
61.
a) Move one stick to make a true equation.
b) Add one stick to make the fraction(1/6) equivalent to one.
c) Move two sticks to make a true equation.
d) Move one stick to make a true equation.
Now There’s An Odd Sum.
62. a) The sum of ten positive odd numbers is 20. Find the largest number which can be used as
an addend in this sum.
{Hint: One of the numbers will be the largest when the other nine as small as possible.}
b) The sum of ten distinct positive numbers is 56. Find the largest number which can be used
as an addend in this sum.
No Empty Promises! I Want The Truth, The Hole Truth, And… .
63. How many cubic inches of dirt are in a hole in the ground that is 1 ft. long by 1 ft. wide by 1
ft. deep?
I Guess I Have To Spell It Out For You.
64. How can 9 horses be used to fill the following row of 10 horse stalls?
{Hint: See the title of the problem.}
This Game’s Just A Pile Of Shi-llings.
65. A game involves a pile of 11 coins and two players who alternately take turns removing 1, 2,
3, 4, or 5 coins from the pile. The player who removes the last coin(s) wins the game. How
many coins should the first player remove in order to guarantee that he can win on his next
turn?
Mr. Sajak, May I Buy A Vowel?
66. It occurs once in a minute, twice in a week, and once in a year. What is it?
Are You A Groupie?
67. Place one set of parentheses on the left side of the equal sign to make the equation true.
32 4 2 8 5 12 4 41
{Hint: Just look at groups of two items.}
If The Glove Don’t Fit, Then I Quit!
68. There are 20 gloves in a drawer: 5 pairs of black gloves(5 left and 5 right), 3 pairs of brown
gloves(3 left and 3 right), and 2 pairs of grey gloves(2 left and 2 right). You will select gloves
from the drawer in the dark, and you may check them only after the selection has been made.
What is the fewest number of gloves you need to select in order to be guaranteed of selecting
at least
a) one matching pair of gloves?(left and right of the same color)
{You could have 5 left black gloves, 3 right brown gloves, 2 left grey gloves, and still not a
matching pair of gloves.}
b) one matching pair of each color?
What’s Left Over?
69. a) If j is an integer, what is the remainder when 2 7 6 5j j is divided by 4?
{Hint: 2 22 7 6 5 12 52 35 4 3 13 8 3j j j j j j .}
b) If x and y are integers, what is the remainder when 5 13 5 3x y is divided by 5?
Making The Most And The Least Of Them.
70. Using each of the four digits 5, 6, 2, and 9 exactly one time,
a) Make two 2-digit numbers that have the largest possible sum.
b) Make two 2-digit numbers that have the smallest possible positive difference.
c) Make two 2-digit numbers that have the largest possible product.
d) Make two 2-digit numbers that have the smallest possible product.
Highs And Lows.
71. If a and b are integers with 30 60a and 60 30b , then find the largest and smallest
possible integer values of the following expressions.
a) a b b) a b c) ab d) a b
On The Mark, Off The Mark, Or Bull’s Eye?
72. In the following figure, the curves are concentric circles with the indicated radii. Which
shaded region has the larger area, the inner circle or the outer ring?
Calculate the area of each region and check your visual estimation ability.
The People Under The Stairs.
73. Three rectangles are connected as in the figure. The first rectangle is 2 by 1; the second
rectangle is 4 by 2; the third rectangle is 8 by 4. A line is drawn from a vertex of the smallest
rectangle to a vertex of the largest rectangle. Find the area of the shaded region.
3 5
4
2 4 8
4
1 2
Grazin’ In The Grass Is A Gas, Baby, Can You Dig It?
74. a) A horse is tethered by a rope to a corner on the outside of a square corral that is 10 feet on
each side. The horse can graze at a distance of 18 feet from the corner of the corral where
the rope is tied. What is the exact total grazing area for the horse?
{Hint:
Express the answer using .}
b) Do the same calculation if the horse can graze at a distance of 22 feet from the corner of the
corral where the rope is tied.
Just Go All The Way Around.
75. The following figure consists of six congruent squares, and it has an area of 294. Find the
perimeter of the figure.
18
18
8
8
How Touching!
76. Four circles, each of which has a diameter of 2 feet, touch as shown. Find the exact area of
the shaded portion.
Hint: The area of the shaded portion would be the area of the square minus the area of the
four circular sectors. Express the answer using .
We use a base 10 number system. A number written with the digits abcd actually represents the
number 3 210 10 10a b c d , where the digits a, b, c, and d can be any of the numbers 0, 1,
2, 3, 4, 5, 6, 7, 8, or 9. If we want to use a base of 2, then abcd would represent the number 3 22 2 2a b c d , where the digits a, b, c, and d can be the numbers 0 or 1.
I’m Thinking Of A Number From 1 To 31.
77. A person is asked to think of a whole number from 1 to 31. The person is shown the following
5 cards and asked to identify the cards that contain the number he is thinking of.
1 3 5 7
9 11 13 15
17 19 21 23
25 27 29 31
2 3 6 7
10 11 14 15
18 19 22 23
26 27 30 31
4 5 6 7
12 13 14 15
20 21 22 23
28 29 30 31
8 9 10 11
12 13 14 15
24 25 26 27
28 29 30 31
16 17 18 19
20 21 22 23
24 25 26 27
28 29 30 31
By adding the numbers in the upper left corners of the identified cards, you can always
determine the person’s number. Use base 2 numbers to explain how the trick works.
Oh Yeah, Well I’m thinking Of A Number From 1 to 63.
78. See if you can extend the previous trick to work for whole numbers from 1 to 63 using 6
cards instead of 5. Complete the numbers on the 6 cards in order for the trick to work.
1 3 5 7 9 11 13 15
17 19 21 23 25 27 29 31
33 35 37 39 41 43 45 47
49 51 53 55 57 59 61 63
2
4
8
16
32
{See problem #77.}
In College Algebra, you learned something called the Remainder Theorem for evaluating
polynomials. It says that if you want to find p c where p x is a polynomial, then you can
divide p x by x c , and the remainder will be p c . You also learned synthetic division,
which gave you an efficient way to divide p x by x c , and therefore an efficient way to
evaluate p c . As an example, suppose you want to find 3p , where 3 22 4 5p x x x x
. To find the remainder when p x is divided by 3x , we’ll use synthetic division:
3 2 1 4 5
6 21 51
2 7 17 56
So 3 56p .
When you are given the base b numeral 1 2 1 0n n nd d d d d , its value is
1
1 1 0
n n
n nd b d b d b d
which is what you get when you find p b for the polynomial
1
1 1 0
n n
n np x d x d x d x d
. So an efficient method of converting the base b numeral
1 2 1 0n n nd d d d d into decimal is to use synthetic division.
1 0
?
n n
n
n
d d db
bd
d
This approach is called Horner’s Method.
Little Jack Horner Sat In A Corner Converting His Base B Numerals.
79. Convert the following numerals into equivalent decimals using Horner’s Method:
a) 21101011 b) 312022110 c) 762134
Getting To Second Base.
80. Find the smallest values for a and b so that 21 25b a .
{Hint: 2 1 2 5 2b a b a .}
The Basis For A Leap Year.
81. Find all bases b, so that 2,431 366b .
{Hint: 3 22 4 3 1 366b b b and 5b .}
Not All Things Must Pass.
82. Forty-one students each took three exams: one in Algebra, one in Biology, and one in
Chemistry. Here are the results
12 failed the Algebra exam
5 failed the Biology exam
8 failed the Chemistry exam
2 failed Algebra and Biology
6 failed Algebra and Chemistry
3 failed Biology and Chemistry
1 failed all three
How many students passed exams in all three subjects?
{Hint: Make a Venn diagram.}
Man Or Woman, We Mean Business.
83. Out of 35 students in a math class, 22 are male, 19 are business majors, 27 are first-year
students, 14 are male business students, 17 are male first-year students, 15 are first-year
students who are business majors, and 11 are male first-year business majors.
a) How many upper class female non-business majors are in the class?
b) How many female business majors are in the class?
The Joy Of Sets.
84. The elements of the set B are all the possible subsets of the set A. Set B has 16 subsets. Find
the number of elements in set A.
Male Female
Business
Non-business
First-year
student
Trains, Planes, And Automobiles.
85. 85 travelers were questioned about the method of transport they used on a particular day.
Each of them used one or more of the methods shown in the Venn diagram. Of those
questioned, 6 traveled by bus and train only, 2 by train and car only, and 7 by bus, train, and
car. The number x who traveled by bus only was equal to the number who traveled by bus
and car only. 35 people used buses, and 25 people used trains. Find:
a) the value of x.
b) the number who traveled by train only.
c) the number who traveled by at least two methods of transport.
d) the number who traveled by car only.
Bus Train
Car U
6
7
2 x
x
The A, B, C, And D Of Education.
86. The results of a survey were the following:
12 students take art, 20 take biology, 20 take chemistry, 8 take drama, 5 take art and biology,
7 take art and chemistry, 4 take art and drama, 16 take biology and chemistry, 4 take biology
and drama, 3 take chemistry and drama, 3 take art, biology, and chemistry, 2 take art, biology,
and drama, 2 take biology, chemistry, and drama, 3 take art, chemistry, and drama, 2 take all
four, 71 take none of the four
a) How many students participated in the survey?
b) How many take exactly one class?
c) How many take exactly two classes?
d) How many take exactly three classes?
e) How many take two or more classes?
Art
Biology Chemistry
Drama
U
The Truth About Cats And Dogs And Birds And Fish.
87. A survey of 136 pet owners resulted in the following information: 49 own fish; 55 own a
bird; 50 own a cat; 68 own a dog; 2 own all four; 11 own only fish; 14 own only a bird; 10
own fish and a bird; 21 own fish and a cat; 26 own a bird and a dog; 27 own a cat and a
dog; 3 own fish, a bird, a cat, but no dog; 1 owns fish, a bird, a dog, but no cat; 9 own fish,
a cat, a dog, but no bird; and 10 own a bird, a cat, a dog, but no fish.
How many of the surveyed pet owners have no fish, no birds, no cats, and no dogs?
Consider the sets A and B inside a universal set U. n U n A B n A B , so we get that
n U n A n B n A B n A B . This rearranges into
n A B n A n B n U n A B , and since 0n A B , it must be that
n A B n A n B n U . This means that the number of elements in the intersection of
A and B is at least n A n B n U , and it also means that if 0n A n B n U , then
it’s possible that 0n A B . This result can be extended to the case of three sets as follows:
n A B C n A B C n A n B C n U n A n B n C n U n U
so 2n A B C n A n B n C n U . It can further be extended to the case of four
sets as follows:
Fish
Bird Cat
Dog
U
2
n A B C D n A B C D n A n B C D n U
n A n B n C n D n U n U
, so
3n A B C D n A n B n C n D n U .
In general, you can show that
1 2 1 2 1k kn A A A n A n A n A k n U .
Also, n A B n A and n A B n B , so min ,n A B n A n B . In general,
you can show that 1 2 1 2min , , ,k kn A A A n A n A n A .
War Is Hell!
88. In a group of 100 war veterans, if 70 have lost an eye, 75 an ear, 80 an arm, and 85 a leg:
a) at least how many have lost all four?
{Hint: See the previous discussion.}
b) at most how many have lost all four?
If You Can’t Work On Transmissions, That’s The Brakes.
89. A car shop has 12 mechanics, of whom 8 can work on transmissions and 7 can work on brakes.
a) What is the minimum number who can do both?
b) What is the maximum number who can do both?
c) What is the minimum number who can do neither?
d) What is the maximum number who can do neither?
{Hint: See the hint for problem #88.}
Hardback Or Paperback Writer?
90. Books were sold at a school book fair. Each book sold was either fiction or nonfiction and
was either hardback or paperback. The chair-person of the book-selling committee can’t
remember exactly how many hardback books of fiction were sold, but he does remember
that
30 books were sold in all
20 hardcover books were sold
15 books of fiction were sold
a) What is the smallest possible number of hardback books of fiction sold?
b) What is the largest possible number of hardback books of fiction sold?
{Hint: See the hint for problem #88.}
It’s All Ancient Egyptian To Me.
91. Addition and subtraction are easy with hieroglyphic numerals: just replace ten symbols with
one symbol of ten times the value. Convert 456 and 557 into hieroglyphic numerals, add
them together, and express the answer in both hieroglyphic and base ten numerals.
Decimal
numeral
Hieroglyphic
symbol
It’s All Ancient Egyptian Cursive To Me.
92. Hieroglyphic numerals were mainly used on stone monuments. For writing on papyrus,
ancient Egyptians used hieratic numerals.
Convert 456 and 557 into hieratic numerals, add them together, and express the answer in
both hieratic and base ten numerals.
Born In Babylonia, Raised In Arizona.
93. The Babylonians used a positional system with base 60. Here is a table of the 59 non-zero
digits.
a) Add the two three-digit Babylonian cuneiform numerals
, and express the answer in both cuneiform and decimal numerals. Assume that the right-
most position is the ones position.
b) There were two versions of cuneiform, Sumerian and Babylonian, as illustrated below
with 4-digit numerals in both versions.
Determine the equivalent decimal numeral.
and
Mama Maya?
94. The Maya initially used a positional system with base 20. Here is a table of the 20 digits.
It was usually written vertically, but sometimes horizontally.
Write the decimal equivalent of the previous Mayan numeral
that is written both vertically and horizontally.
020
120
220
320
420
