- Handley High School has exactly 1000 lockers and exactly 1000 students.
On the first day of school, the students meet outside the building and agree
on the following plan:
The first student will enter the school and open all the lockers.
The second student will enter the school and close every locker with an even number
(2, 4, 6, 8, ...).
The third student will then reverse every third locker. That is, if the locker is open,
she will close it; if the locker is closed, she will open it.
The fourth student will then reverse every fourth locker, and so on until all 1,000 students
have entered the building and reversed the proper lockers.
Which lockers will finally remain open?
- The Cotters are throwing a party.
The first time the door bell rings, 1 guest enters.
On the second ring, 3 guests enter.
On the third ring, 5 guests enter, and so on.
How many guests will enter on the fiftieth ring?
How many total guests will be present after the fiftieth ring?
- How many total gifts are given in the song, "The Twelve Days of Christmas?"
- There are 8 people in a room.
Each person shakes hands with each of the other people once and only once.
How many handshakes take place?
- How many squares are there on a standard checkerboard?
(The answer is not 64).
- Find all rectangles with integral sides whose area and perimeter are numerically equal.
- A student has some pigs and some chickens.
He finds that together they have 70 heads and 200 legs.
How many pigs and how many chickens does he have?
- If you can run the "600 yard run" in 1 minute, 40 seconds,
how fast is that in miles per hour?
- Mike and Barbara are both good chess players.
They have completed five games and each has won the same number of games.
There were no ties.
How did this happen?
- You are offered a job that pays 1 cent on the first day, 2 cents the second day,
4 cents the third day, 8 cents the fourth day, and so on for 30 days.
How much will be paid on the last day, and what is the total for all 30 days?
- Take a sheet of paper and tear it in half.
Place the two pieces of paper on top of each other, and tear them in half.
This gives 4 pieces of paper.
Place them on top of one another and tear them in half again.
If you were to continue this process through 20 such tears, how many sheets will be in the pile?
If each sheet is .001 inches thick, how high is the pile of paper?
- There are 50 teams entered in a single elimination tournament.
Winners play until only one team is left.
What is the total number of games that must be played?
Do not count byes as games played.
- A ship at point A is 50 yards directly north of point C on shore.
Point D, also on the shore and due east of point C, is 130 yards from point C.
Point B, a lighthouse, is due north of point D and 80 yards from point D.
The ship must touch the shoreline and then sail to the lighthouse.
Find the location of point X on the shore so that the path from A to X to B will
be a minimum.
What is the minimum distance (AX + XB)?
- What is the smallest number of coins that you need so that you can give exact change
for any amount from 1 cent to 99 cents?
- Three missionaries and three cannibals wish to cross a river.
There is a boat that can carry up to three people, and both the missionaries
and the cannibals can row the boat.
However, the cannibals can never outnumber the missionaries (either in the boat
or on shore) or the cannibals will feast upon the missionaries.
What is the smallest number of trips needed to make the crossing?
- When I arrange the members of a marching band in rows of 2, 3, or 4, there is always
1 person left over.
However, when I arrange them in rows of 5, the rows are all even.
What is the minimum number of people in the marching band?
- The sum of two numbers is 2,
and the product of these same two numbers is 3.
Determine the sum of their reciprocals.
- During a recent census, a man told the census taker that he had 3 children.
When asked their ages, he replied, "The product of their ages is 72."
"The sum of their ages is the same as my house number."
The census taker ran to the door and looked at the house number.
"I still can't tell," she complained.
"Oh, that's right. I forgot to tell you that the oldest one likes apple pie."
The census taker promptly wrote down the ages of the three children.
How old are they?
- When the math teacher returned from his fishing trip,
his students inquired as to the length of his prize catch.
The teacher answered,
"The head measured 9 inches."
"The tail was a slong as the head and half the body."
"The body was as long as the head and tail."
How long was the math teacher's prize fish?
- If x represents a number on a 12-hour clock and 4 + x = 6 - x,
determine the value of x.
(Find all answers between 0 and 12 inclusive).
- If ping pong balls come in packages of either 8 or 15 to a box,
what is the largest number of ping pong balls that can be ordered
that will require breaking a box to fill the order?
- Using each of the ten digits once,
can you find two five-digit numbers with the largest product?
- Three sailors plan to divide a pile of coconuts among themselves in the morning.
During the night, one of the sailors decides to take his share (1/3 of the pile).
A short while later, the second sailor awoke and took 1/3 of the remaining coconuts.
Still later, the third sailor awoke and took 1/3 of the remaining coconuts.
In the morning, there were 8 coconuts in the pile.
How many coconuts were there originally?
- In the first week of the baseball season, James gets 5 hits out of 11 at bats.
In the first week of the baseball season, Jackson gets 3 hits out of 7 at bats.
Who had the better batting average for the first week?
In the second week of the baseball season, James gets 6 hits out of 9 at bats.
In the second week of the baseball season, Jackson gets 9 hits out of 14 at bats.
Who had the better batting average for the second week?
Who had the better batting average for the total of the first two weeks?
- A string is tied tightly around the Earth's equator.
An additional 10 feet is spliced in.
If the slack is evenly distributed, how high off the Earth's surface will the string be?
- Suppose you are 6 feet tall and you walk around the Earth's equator.
How much farther does your head travel than your feet?
- In how many ways can MATHEMATICS be spelled going from one letter to an adjacent letter?
- An automobile went from New York to Chicago at an average rate of 30 mph.
How fast would it have to come back to make the average speed for the round trip 60 mph?
- A car went up a hill at a speed of 10 mph, and came back downhill at a speed of 20 mph.
What was the average speed for the round trip?
- Determine the value of the fraction below:
- A commuter is in the habit of arriving at his suburban
station each evening at exactly 6:00 PM.
His wife is waiting for him with the car; she, too, arrives
at exactly 6:00 PM.
She never varies her route or her rate of speed.
One day, the commuter takes an earlier train, arriving at
the station at 5:00 PM.
He decides not to call his wife, but begins to walk toward
home along the route she always takes.
They meet somewhere along the route; he gets into the car;
and they drive home.
They arrive home 10 minutes earlier than usual.
How long had the husband been walking when he was picked
up by his wife?
- Edie and Dave were talking when they saw three people coming toward them.
"I wonder how old they are," said Edie.
Dave replied, "I know them!
The product of their ages is 2,450 and the sum of their ages is twice your age."
"That's all well and good," said Edie, "but I need more information."
"Oh yes," said Dave.
"Well, I am older than any of the three."
"Now, I can figure their ages," said Edie.
How old are the three?
- Two bike riders, Jeff and Nancy, are 25 miles apart,
riding toward each other at speeds of 15 mph and 10 mph, respectively.
A fly starts from Jeff and flies toward Nancy and then back to Jeff again and so on.
The fly continues flying back and forth at a constant rate of 40 mph
until the bicycle riders collide and crush the fly.
How far has the fly traveled?
Does it really matter?
- There are 32 teams in the National football League.
To conduct their annual draft,
teams in each city must have a direct phone line to each of the other cities.
How many direct phone lines are required?
- In a game called crossball, a team can score either 3 points or 7 points.
Which scores can a team not make?
- A man starts out at home.
He goes to a corner, turns 90 degrees left,
goes to another corner, turns 90 degrees left,
goes to another corner, and turns 90 degrees left.
When he gets back home, he meets two masked men.
Who are they?
- A man goes 1 mile South, 1 mile East,
and 1 mile North, arriving back at his home.
He sees a bear.
What color is it?
- A man headed north and walked 10 miles in a straight line, then stopped.
A little while later, he headed north again,
walked 10 more milesin a straight line -- and found himself back where he started!
How is that possible?
- A well is 10 feet deep.
A frog climbs up 5 feet during the day, but falls back 4 feet at night.
Assuming that the frog starts at the bottom of the well,
how many days does it take the frog to get to the top?
- For security reasons, many hotels use security codes on their keys.
Below is a list of room numbers and their accompanying codes.
Can you decipher the code and determine the room number on the last line?
Hint: There is no room 555.
Room Number
Code Number
623
487
509
501
1487
9623
?
365
- Henry Wadsworth Longfellow introduced the following into his novel Kavanaugh:
When the stem of a water lily is vertical,
the blossom is 10 centimeters above the surface of a lake.
If you pull the lily to one side keeping the stem straight,
the blossom touches the water at a spot 21 centimeters
from where the stem formerly cut the surface.
How deep is the water?
- Read the statements in the box.
Is statement 3 true or false?
- There are three numbered statements in this box.
- Two of the numbered statements are not true.
- Using only these three statements, you can figure out the answer to the question.
- Paul, John, and George are three rock stars.
One plays a guitar, one plays the drums, and one plays the piano,
but not necessarily in that order.
The drummer tried to hire the guitarist for a recording session,
but was told that he was out of town doing shows with the pianist.
The drummer admired the work of both musicians.
The pianist earns more money than the drummer.
Paul earns less than John.
George has never heard of John.
What instrument does each of the rock stars play?
- Here is a complete list of states in the order they appear on a trip taken by countless people every year.
Can you identify the nature of this trip?
VERMONT
CONNECTICUT
VIRGINIA
TENNESSEE
NEW YORK
KENTUCKY
INDIANA
ILLINOIS
NORTH CAROLINA
PENNSYLVANIA
- A census worker asks a lady for the ages of her three children.
The lady replies that the product of their ages is 36,
and the sum of their ages is the same as the next address to the north.
After looking at that address, the census worker returns
and says that she needs more information.
The lady gives her one final clue and says,
"The oldest one likes chocolate pudding."
With that, the census worker was able to figure out the ages.
What are the ages of the children,
and what is that address next door?
- What is the minimum number of pitches possible for a pitcher
to make in a major league baseball game, assuming he pitches the entire game
and it is not called prior to completion?
- Four teachers bought an old boat for $60.
The first teacher paid one-half of the sum of the amounts paid by the other teachers.
The second teacher paid one-third of the sum of the amounts paid by the other teachers.
The third teacher paid one-fourth of the sum of the amounts paid by the other teachers.
How much did the fourth teacher pay?
- In a menagerie, there are birds and beasts.
Determine the number of each given the fact that the lot has 36 heads and 100 feet.
- How many problems must a student get right to score 100 on a 120 question test
if the teacher gives one point for each correct answer but subtracts 1/4 point for each wrong answer?
- A man divided his estate among his three children in the following manner:
The oldest child received $3,600.
The second child received 20% more than the oldest child and 25% more than the youngest.
How much money was bequeathed to the youngest?
- To number the pages in a math book, the printer used 2,989 digits.
How many pages are in the book?
- A man traveled at 55 mph for the first half of his trip and 70 mph on the return trip.
If his total time round-trip is 45 hours, what is the total distance of the trip?
- Henry Wadsworth Longfellow (1807-1882) was an accomplished mathematician who created problems in poetic settings.
Can you find the number of water lilies in this problem created by Longfellow?
One-third of a collection of beautiful water lilies is offered to Mahadev,
one-fifth to Huri, one-sixth to the Sun, one-fourth to Devi,
and six which remain are presented to the spiritual teacher.
- The poet Longfellow, in his novel, Kavanaugh, introduced several clever mathematical
problems from an ancient Sanskrit work. One follows...
"If one fifth of a bee hive of bees flew to the ladamba flower,
on third flew to the slandbara, three times the difference of these two numbers flew to an arbor,
and one bee continued to fly about attracted on each side by fragrant ketaki and the malati,
what was the number of bees?"
On the first day of school, the students meet outside the building and agree on the following plan:
The first student will enter the school and open all the lockers.
The second student will enter the school and close every locker with an even number (2, 4, 6, 8, ...).
The third student will then reverse every third locker. That is, if the locker is open, she will close it; if the locker is closed, she will open it.
The fourth student will then reverse every fourth locker, and so on until all 1,000 students have entered the building and reversed the proper lockers.
Which lockers will finally remain open?
The first time the door bell rings, 1 guest enters.
On the second ring, 3 guests enter.
On the third ring, 5 guests enter, and so on.
How many guests will enter on the fiftieth ring?
How many total guests will be present after the fiftieth ring?
Each person shakes hands with each of the other people once and only once.
How many handshakes take place?
(The answer is not 64).
He finds that together they have 70 heads and 200 legs.
How many pigs and how many chickens does he have?
how fast is that in miles per hour?
They have completed five games and each has won the same number of games.
There were no ties.
How did this happen?
How much will be paid on the last day, and what is the total for all 30 days?
Place the two pieces of paper on top of each other, and tear them in half.
This gives 4 pieces of paper.
Place them on top of one another and tear them in half again.
If you were to continue this process through 20 such tears, how many sheets will be in the pile?
If each sheet is .001 inches thick, how high is the pile of paper?
Winners play until only one team is left.
What is the total number of games that must be played?
Do not count byes as games played.
Point D, also on the shore and due east of point C, is 130 yards from point C.
Point B, a lighthouse, is due north of point D and 80 yards from point D.
The ship must touch the shoreline and then sail to the lighthouse.
Find the location of point X on the shore so that the path from A to X to B will be a minimum.
What is the minimum distance (AX + XB)?
There is a boat that can carry up to three people, and both the missionaries and the cannibals can row the boat.
However, the cannibals can never outnumber the missionaries (either in the boat or on shore) or the cannibals will feast upon the missionaries.
What is the smallest number of trips needed to make the crossing?
However, when I arrange them in rows of 5, the rows are all even.
What is the minimum number of people in the marching band?
and the product of these same two numbers is 3.
Determine the sum of their reciprocals.
When asked their ages, he replied, "The product of their ages is 72."
"The sum of their ages is the same as my house number."
The census taker ran to the door and looked at the house number.
"I still can't tell," she complained.
"Oh, that's right. I forgot to tell you that the oldest one likes apple pie."
The census taker promptly wrote down the ages of the three children.
How old are they?
his students inquired as to the length of his prize catch.
The teacher answered,
"The head measured 9 inches."
"The tail was a slong as the head and half the body."
"The body was as long as the head and tail."
How long was the math teacher's prize fish?
determine the value of x.
(Find all answers between 0 and 12 inclusive).
what is the largest number of ping pong balls that can be ordered
that will require breaking a box to fill the order?
can you find two five-digit numbers with the largest product?
During the night, one of the sailors decides to take his share (1/3 of the pile).
A short while later, the second sailor awoke and took 1/3 of the remaining coconuts.
Still later, the third sailor awoke and took 1/3 of the remaining coconuts.
In the morning, there were 8 coconuts in the pile.
How many coconuts were there originally?
In the first week of the baseball season, Jackson gets 3 hits out of 7 at bats.
Who had the better batting average for the first week?
In the second week of the baseball season, James gets 6 hits out of 9 at bats.
In the second week of the baseball season, Jackson gets 9 hits out of 14 at bats.
Who had the better batting average for the second week?
Who had the better batting average for the total of the first two weeks?
An additional 10 feet is spliced in.
If the slack is evenly distributed, how high off the Earth's surface will the string be?
How much farther does your head travel than your feet?
How fast would it have to come back to make the average speed for the round trip 60 mph?
What was the average speed for the round trip?
station each evening at exactly 6:00 PM.
His wife is waiting for him with the car; she, too, arrives
at exactly 6:00 PM.
She never varies her route or her rate of speed.
One day, the commuter takes an earlier train, arriving at
the station at 5:00 PM.
He decides not to call his wife, but begins to walk toward
home along the route she always takes.
They meet somewhere along the route; he gets into the car;
and they drive home.
They arrive home 10 minutes earlier than usual.
How long had the husband been walking when he was picked
up by his wife?
"I wonder how old they are," said Edie.
Dave replied, "I know them!
The product of their ages is 2,450 and the sum of their ages is twice your age."
"That's all well and good," said Edie, "but I need more information."
"Oh yes," said Dave.
"Well, I am older than any of the three."
"Now, I can figure their ages," said Edie.
How old are the three?
riding toward each other at speeds of 15 mph and 10 mph, respectively.
A fly starts from Jeff and flies toward Nancy and then back to Jeff again and so on.
The fly continues flying back and forth at a constant rate of 40 mph
until the bicycle riders collide and crush the fly.
How far has the fly traveled?
Does it really matter?
To conduct their annual draft,
teams in each city must have a direct phone line to each of the other cities.
How many direct phone lines are required?
Which scores can a team not make?
He goes to a corner, turns 90 degrees left,
goes to another corner, turns 90 degrees left,
goes to another corner, and turns 90 degrees left.
When he gets back home, he meets two masked men.
Who are they?
and 1 mile North, arriving back at his home.
He sees a bear.
What color is it?
A little while later, he headed north again,
walked 10 more milesin a straight line -- and found himself back where he started!
How is that possible?
A frog climbs up 5 feet during the day, but falls back 4 feet at night.
Assuming that the frog starts at the bottom of the well,
how many days does it take the frog to get to the top?
Below is a list of room numbers and their accompanying codes.
Can you decipher the code and determine the room number on the last line?
Hint: There is no room 555.
| Room Number | Code Number |
|---|---|
| 623 | 487 |
| 509 | 501 |
| 1487 | 9623 |
| ? | 365 |
When the stem of a water lily is vertical,
the blossom is 10 centimeters above the surface of a lake.
If you pull the lily to one side keeping the stem straight,
the blossom touches the water at a spot 21 centimeters
from where the stem formerly cut the surface.
How deep is the water?
Is statement 3 true or false?
|
One plays a guitar, one plays the drums, and one plays the piano,
but not necessarily in that order.
The drummer tried to hire the guitarist for a recording session,
but was told that he was out of town doing shows with the pianist.
The drummer admired the work of both musicians.
The pianist earns more money than the drummer.
Paul earns less than John.
George has never heard of John.
What instrument does each of the rock stars play?
Can you identify the nature of this trip?
VERMONT
CONNECTICUT
VIRGINIA
TENNESSEE
NEW YORK
KENTUCKY
INDIANA
ILLINOIS
NORTH CAROLINA
PENNSYLVANIA
The lady replies that the product of their ages is 36,
and the sum of their ages is the same as the next address to the north.
After looking at that address, the census worker returns
and says that she needs more information.
The lady gives her one final clue and says,
"The oldest one likes chocolate pudding."
With that, the census worker was able to figure out the ages.
What are the ages of the children,
and what is that address next door?
to make in a major league baseball game, assuming he pitches the entire game
and it is not called prior to completion?
The first teacher paid one-half of the sum of the amounts paid by the other teachers.
The second teacher paid one-third of the sum of the amounts paid by the other teachers.
The third teacher paid one-fourth of the sum of the amounts paid by the other teachers.
How much did the fourth teacher pay?
Determine the number of each given the fact that the lot has 36 heads and 100 feet.
if the teacher gives one point for each correct answer but subtracts 1/4 point for each wrong answer?
The oldest child received $3,600.
The second child received 20% more than the oldest child and 25% more than the youngest.
How much money was bequeathed to the youngest?
How many pages are in the book?
If his total time round-trip is 45 hours, what is the total distance of the trip?
One-third of a collection of beautiful water lilies is offered to Mahadev,
one-fifth to Huri, one-sixth to the Sun, one-fourth to Devi,
and six which remain are presented to the spiritual teacher.
problems from an ancient Sanskrit work. One follows...
"If one fifth of a bee hive of bees flew to the ladamba flower,
on third flew to the slandbara, three times the difference of these two numbers flew to an arbor,
and one bee continued to fly about attracted on each side by fragrant ketaki and the malati,
what was the number of bees?"
Answers
--------------------
- 31 lockers will remain open.
Construct a table. The perfect squares 1, 4, 9, 16, 25, ..., 961
will be the open lockers.
- 99 guests enter on the 50th ring, and there are 2,500 guests present.
Use a table for the ring, the number of guests entering,
and the total guests, and then look for a pattern.
Ring #
Guests
Total
1
1
1
2
3
4
3
5
9
4
7
16
5
9
25
6
11
36
n
2n-1
n2
50
99
2500
-
There are 364 gifts. Click here for an explanation.
- There are 28 handshakes.
Draw an octagon.
Use the formula (n(n-3))/2, which gives you the total number of diagonals,
and then add n to it (for the number of sides of the polygon).
- There are 204 squares.
Use a chart:
Dimensions of the square
Number of Squares
1 x 1
64
2 x 2
49
3 x 3
36
4 x 4
25
5 x 5
16
6 x 6
9
7 x 7
4
8 x 8
1
- There are 2 rectangles: 4 x 4 and 6 x 3.
Solve 2a + 2b = ab.
- He has 30 pigs and 40 chickens.
Solve: 1p + 1 c = 70 and 4p + 2c = 200 simultaneously.
- 12.2 mph.
- They didn't play each other.
- Pay on the last day is over $5 million, and the total is over $10 million.
Click here to see the computation.
- There would be 220 or 1,048,576 sheets of paper,
and the pile would be 1,048 inches or 87 feet tall.
Click here for more information.
- 49 games.
There is one winner, so there must be 49 losers, and hence, 49 games played.
- Point X must be 50 yards from C and 80 yards from D.
The minimum distance is 183.8 yards.
- You need 9 coins.
4 pennies, 1 nickel, 2 dimes, 1 quarter, 1 half dollar, or
4 pennies, 2 nickels, 1 dime, 1 quarter, 1 half dollar.
- Five crossings are needed.
(1) Send 3 cannibals across.
(2) Then 1 cannibal comes back.
(3) Then 3 missionaries row across.
(4) Then 1 missionary comes back.
(5) The missionary and cannibal row across.
- The smallest number is 25.
The least common multiple of 2, 3, and 4 is 12.
Add 1. The sequence is 13, 25, 37, 49, ...
- The sum of the reciprocals is 2/3.
- The ages of the three children are 3, 3, and 8.
First, list all 12 sets of factor triples that equal 72:
1 x 1 x 72 sum = 74
1 x 2 x 36 sum = 39
1 x 3 x 24 sum = 28
1 x 4 x 18 sum = 23
1 x 6 x 12 sum = 19
1 x 8 x 9 sum = 18
2 x 2 x 18 sum = 22
2 x 3 x 12 sum = 17
2 x 4 x 9 sum = 15
2 x 6 x 6 sum = 14
3 x 3 x 8 sum = 14
3 x 4 x 6 sum = 13
There are two sums that equal 14, and that is why the census taker couldn't tell.
So, the additional clue about apple pie tells the census taker that there is one oldest child.
Hence, the ages are 3, 3, and 8, and not 2, 6, and 6.
The house number is 14.
- The fish was 72 inches long.
Let b = length of body.
Let t = length of tail.
then t = 9 + 1/2 b and b = 9 + t.
Solving simultaneously, b = 36" and t = 27".
- x = 1 and 7.
Solving the equation gives you x = 1, but remember that 11 = -1 in base 12.
- The largest number is 97.
- The two numbers are 96,420 and 87,531 with a product of 8,439,739,020.
Use two principles:
(1) The largest digits go to the left and
(2) the product of 2 numbers will be maximized if their difference is as small as possible.
- There were 27 coconuts in the original pile.
The first sailor took 9 coconuts, leaving 18.
The second sailor took 6 coconuts, leaving 12.
The third sailor took 4 coconuts, leaving 8.
- James had the better average for the first week (5/11 = .454 to 3/7 = .428).
James had the better average for the second week (6/9 = .667 to 9/14 = .642).
However, even though James had a better average for each week,
Jackson had the better overall average (12/21 = .571 to 11/20 = .550)!
- The answer is 1.59 feet off the ground!
- Your head travels 37.7 feet farther than your feet.
- There are 252 different ways.
Use an adaptation of Pascal's triangle:
- It is not possible to average 60 mph no matter how fast you come back.
If the distance from New York to Chicago is 900 miles,
then it would take 30 hours to get there at 30 mph.
To average 60 mph over the whole trip of 1,800 miles,
you would have to make the trip in 30 hours,
but you already used 30 hours going there,
so you would have to make the return trip in zero hours.
- The average speed is 13.333 mph.
Let the distance up the hill be 20 miles.
Then it took 2 hours to go up and 1 hour to come back down.
So, the average speed is found by dividing total distance (40 miles) by total time (3 hours).
- The value is 1.
Use reciprocal identities or sin x csc x = 1, cos x sec x = 1, and tan x cot x = 1.
- The commuter had been walking 55 minutes.
From the point of view of the wife:
The car has been going 5 minutes less in each direction.
The husband had been walking 55 minutes.
- They are 49, 10, and 5 years old.
Begin by lookig at all the factors whose product is 2,450.
Then look at the sums of those factors.
Factors of 2,450
Sum of Factors
1-49-50
100
2-25-49
76
2-35-35
72
5-5-98
108
5-7-70
102
5-10-49
64
5-14-35
54
7-7-50
64
7-14-25
46
7-10-35
52
Since Edie cannot figure out the ages at first,
it must be because there are two sums equal to twice her age.
5 + 10 +49 = 64 and 7 + 7 + 50 = 64, so Edie must be 32.
When Dave gives the final clue, it reveals that he is 50,
and therefore, the three ages that work are 5, 10, and 49.
- The fly travels 40 miles.
The bike riders will collide in one hour
since they are going toward each other at a combined 25 mph.
- The answer is 496 phone lines.
Use this formula: n + n(n - 3)/2, where n is the number of teams.
Visualize an n-sided polygon, with n(n - 3)/2 diagonals.
- A team cannot score 1, 2, 4, 5, 8, and 11 poiints.
- The catcher and the umpire.
- The bear is white.
He must have started from the North Pole.
- The man started at a point less than 10 miles from the North Pole.
First, he headed north and walked 10 miles,
passing over the pole and continuing for some distance.
Then he turned around, headed north again,
and walked 10 more miles.
- It will take 6 days.
At the end of the first 5 days, he will have climbed 5 feet.
Then on the 6th day, he will climb to 10 feet and be out!
- The last room number is 745.
Subtract each individual digit in the room number from 10 to crack the code.
- It is 341/20 = 17.05 cm.
- Statement 3 is true.
Statement 1 is obviously true.
So, statement 2 must be false --
if it were true, it would be a contradiction
(two statements must not be false).
Therefore, statement 3 must be true.
- Paul plays the guitar,
John plays the drums,
George plays the piano.
- The list contains the states appearing in the names of avenues on a Monopoly Board
in the order they are encountered traveling around the board.
- The children's ages are 2, 2, and 9, and the address is 13.
The following are the only combinations of three ages whose product is 36:
1x1x36 2x2x9 1x3x12 1x2x18
1x6x6 2x3x6 1x4x9 3x3x4
Except for two of the combinations, their sums are all different,
so the census worker would have been able to determine the ages of
the children if the address next door had been any of those six different ones.
As she needed more information, the address must have been 13,
a total shared by two combinations: 1, 6, 6, and 2, 2, 9.
So, when the mother indicated that there was an oldest child,
she eliminated the first combination, which had two "oldest,"
leaving only 2, 2, 9 as the answer.
- The minimum number of pitches is 25.
He is the losing pitcher and he is playing away from home.
He retires the first 24 batters on 1 pitch each (3 pitches per inning)
and then the first batter up in the ninth inning hits a homerun to win the game, 1-0.
- The answer is $13.
- There are 22 birds and 14 beasts.
Let A = number of birds and B = the number of beasts.
Then A + B = 36 and 2A + 4B = 100
- The student must get 104 questions correct with 16 wrong answers
(104 - 1/4(16) = 100).
- The third child would get $3,456.
- There are 1,024 pages in the book.
A book of 999 pages needs 1(9) + 2(90) + 3(900) or 2,889 digits.
A book of x pages needs 2889 + 4(x - 999) = 2989
Therefore, x = 1024.
- The total trip is 2,772 miles (1,386 miles each way).
Let x = time at which he travels 55 mph, so the distance is 55x.
Then 45 - x is the number of hours at which he traveled 70 mph.
So the return distance is 70(45-x).
Set 70(45-x) = 55x and solve for x.
x = 25.2 hours.
- There are 120 lilies.
Let x = number of lilies.
Then, 1/3 x + 1/5 x + 1/6 x + 1/4 x + 6 = x
- There were 15 bees.
Construct a table. The perfect squares 1, 4, 9, 16, 25, ..., 961 will be the open lockers.
Use a table for the ring, the number of guests entering,
and the total guests, and then look for a pattern.
| Ring # | Guests | Total |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 3 | 4 |
| 3 | 5 | 9 |
| 4 | 7 | 16 |
| 5 | 9 | 25 |
| 6 | 11 | 36 |
| n | 2n-1 | n2 |
| 50 | 99 | 2500 |
Draw an octagon.
Use the formula (n(n-3))/2, which gives you the total number of diagonals, and then add n to it (for the number of sides of the polygon).
Use a chart:
| Dimensions of the square | Number of Squares | |
|---|---|---|
| 1 x 1 | 64 | |
| 2 x 2 | 49 | |
| 3 x 3 | 36 | |
| 4 x 4 | 25 | |
| 5 x 5 | 16 | |
| 6 x 6 | 9 | |
| 7 x 7 | 4 | |
| 8 x 8 | 1 |
Solve 2a + 2b = ab.
Solve: 1p + 1 c = 70 and 4p + 2c = 200 simultaneously.
Click here to see the computation.
and the pile would be 1,048 inches or 87 feet tall.
Click here for more information.
There is one winner, so there must be 49 losers, and hence, 49 games played.
The minimum distance is 183.8 yards.
4 pennies, 1 nickel, 2 dimes, 1 quarter, 1 half dollar, or
4 pennies, 2 nickels, 1 dime, 1 quarter, 1 half dollar.
(1) Send 3 cannibals across.
(2) Then 1 cannibal comes back.
(3) Then 3 missionaries row across.
(4) Then 1 missionary comes back.
(5) The missionary and cannibal row across.
The least common multiple of 2, 3, and 4 is 12.
Add 1. The sequence is 13, 25, 37, 49, ...
First, list all 12 sets of factor triples that equal 72:
1 x 1 x 72 sum = 74
1 x 2 x 36 sum = 39
1 x 3 x 24 sum = 28
1 x 4 x 18 sum = 23
1 x 6 x 12 sum = 19
1 x 8 x 9 sum = 18
2 x 2 x 18 sum = 22
2 x 3 x 12 sum = 17
2 x 4 x 9 sum = 15
2 x 6 x 6 sum = 14
3 x 3 x 8 sum = 14
3 x 4 x 6 sum = 13
There are two sums that equal 14, and that is why the census taker couldn't tell.
So, the additional clue about apple pie tells the census taker that there is one oldest child.
Hence, the ages are 3, 3, and 8, and not 2, 6, and 6.
The house number is 14.
Let b = length of body.
Let t = length of tail.
then t = 9 + 1/2 b and b = 9 + t.
Solving simultaneously, b = 36" and t = 27".
Solving the equation gives you x = 1, but remember that 11 = -1 in base 12.
Use two principles:
(1) The largest digits go to the left and
(2) the product of 2 numbers will be maximized if their difference is as small as possible.
The first sailor took 9 coconuts, leaving 18.
The second sailor took 6 coconuts, leaving 12.
The third sailor took 4 coconuts, leaving 8.
James had the better average for the second week (6/9 = .667 to 9/14 = .642).
However, even though James had a better average for each week,
Jackson had the better overall average (12/21 = .571 to 11/20 = .550)!
Use an adaptation of Pascal's triangle:
If the distance from New York to Chicago is 900 miles,
then it would take 30 hours to get there at 30 mph.
To average 60 mph over the whole trip of 1,800 miles,
you would have to make the trip in 30 hours,
but you already used 30 hours going there,
so you would have to make the return trip in zero hours.
Let the distance up the hill be 20 miles.
Then it took 2 hours to go up and 1 hour to come back down.
So, the average speed is found by dividing total distance (40 miles) by total time (3 hours).
Use reciprocal identities or sin x csc x = 1, cos x sec x = 1, and tan x cot x = 1.
From the point of view of the wife:
The car has been going 5 minutes less in each direction.
The husband had been walking 55 minutes.
Begin by lookig at all the factors whose product is 2,450.
Then look at the sums of those factors.
| Factors of 2,450 | Sum of Factors |
|---|---|
| 1-49-50 | 100 |
| 2-25-49 | 76 |
| 2-35-35 | 72 |
| 5-5-98 | 108 |
| 5-7-70 | 102 |
| 5-10-49 | 64 |
| 5-14-35 | 54 |
| 7-7-50 | 64 |
| 7-14-25 | 46 |
| 7-10-35 | 52 |
Since Edie cannot figure out the ages at first,
it must be because there are two sums equal to twice her age.
5 + 10 +49 = 64 and 7 + 7 + 50 = 64, so Edie must be 32.
When Dave gives the final clue, it reveals that he is 50,
and therefore, the three ages that work are 5, 10, and 49.
The bike riders will collide in one hour
since they are going toward each other at a combined 25 mph.
Use this formula: n + n(n - 3)/2, where n is the number of teams.
Visualize an n-sided polygon, with n(n - 3)/2 diagonals.
He must have started from the North Pole.
First, he headed north and walked 10 miles,
passing over the pole and continuing for some distance.
Then he turned around, headed north again,
and walked 10 more miles.
At the end of the first 5 days, he will have climbed 5 feet.
Then on the 6th day, he will climb to 10 feet and be out!
Subtract each individual digit in the room number from 10 to crack the code.
Statement 1 is obviously true.
So, statement 2 must be false --
if it were true, it would be a contradiction
(two statements must not be false).
Therefore, statement 3 must be true.
John plays the drums,
George plays the piano.
in the order they are encountered traveling around the board.
The following are the only combinations of three ages whose product is 36:
1x1x36 2x2x9 1x3x12 1x2x18
1x6x6 2x3x6 1x4x9 3x3x4
Except for two of the combinations, their sums are all different,
so the census worker would have been able to determine the ages of
the children if the address next door had been any of those six different ones.
As she needed more information, the address must have been 13,
a total shared by two combinations: 1, 6, 6, and 2, 2, 9.
So, when the mother indicated that there was an oldest child,
she eliminated the first combination, which had two "oldest,"
leaving only 2, 2, 9 as the answer.
He is the losing pitcher and he is playing away from home.
He retires the first 24 batters on 1 pitch each (3 pitches per inning)
and then the first batter up in the ninth inning hits a homerun to win the game, 1-0.
Let A = number of birds and B = the number of beasts.
Then A + B = 36 and 2A + 4B = 100
(104 - 1/4(16) = 100).
A book of 999 pages needs 1(9) + 2(90) + 3(900) or 2,889 digits.
A book of x pages needs 2889 + 4(x - 999) = 2989
Therefore, x = 1024.
Let x = time at which he travels 55 mph, so the distance is 55x.
Then 45 - x is the number of hours at which he traveled 70 mph.
So the return distance is 70(45-x).
Set 70(45-x) = 55x and solve for x.
x = 25.2 hours.
Let x = number of lilies.
Then, 1/3 x + 1/5 x + 1/6 x + 1/4 x + 6 = x