Puzzle Page

Interactive Puzzles
Puzzle	Web Address
Circle the Cat	http://www.members.shaw.ca/gf3/circle-the-cat.html
Know Your States	http://jimspages.com/States.htm

Algebraic Puzzles
Puzzle	Date		Puzzle	Date		Puzzle	Date		Puzzle	Date
Puzzle 109	1/16/95		Puzzle 111	2/13/95		Puzzle 112	2/20/95		Puzzle 114	3/20/95
Puzzle 119	6/5/95		Puzzle 122	7/31/95		Puzzle 123	8/14/95		Puzzle 124	9/11/95
Puzzle 126	10/9/95		Puzzle 128	11/13/95		Puzzle 155	1/20/97		Puzzle 167	7/07/97
Puzzle 307	7/07/03		Puzzle 390	1/15/07		Puzzle 391	1/29/07		Puzzle 392	2/19/07
Puzzle 419	4/28/08		Puzzle 420	5/12/08		Puzzle 427	9/15/08		Puzzle 429	10/13/08
Puzzle 431	11/10/08		Puzzle 433	1/12/09		Puzzle 434	1/26/09		Puzzle 216	6/27/99

Geometric Puzzles
Puzzle 118	5/15/95		Puzzle 121	7/17/95		Puzzle 261	7/30/01		Puzzle 413	7/17/07
Puzzle 414	1/14/08		Puzzle 415	1/28/08		Puzzle 416	3/10/08		Puzzle 421	6/09/08
Puzzle 422	6/23/08		Puzzle 424	7/28/08		Puzzle 425	8/11/08		Puzzle 426	9/04/08

Spatial Puzzles
Puzzle 110	1/30/95		Puzzle 115	4/03/95		Puzzle 144	7/15/96		Puzzle 154	12/09/96
Puzzle 430	10/27/08

Logic Puzzles
Puzzle 113	4/03/95		Puzzle 116	4/17/95		Puzzle 120	6/19/95		Puzzle 125	9/25/95
Puzzle 127	10/23/95		Puzzle 418	4/14/08		Puzzle 423	7/14/08		Puzzle 428	9/29/08
Puzzle 250	2/26/01

Arithmetic Puzzles
Puzzle 117	5/01/95		Puzzle 130	12/04/95		Puzzle 145	7/29/96		Puzzle 258	6/18/01
Puzzle 432	12/08/08		Puzzle 435	2/11/09		Puzzle 436	2/11/09		Puzzle 437	3/30/09
Puzzle 253	4/02/01		Puzzle 207	Unknown

Calculus Puzzles
Puzzle 417	2/15/08

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Puzzle 109

Using four numeral 4's, no more, no less, and any combinations of mathematical operations, eg. "x", "/", "+", "-", "!", "√", "(",")", generate expressions that equal the cardinal numbers 1, 2, 3, 4, etc. up to at least 30. For example: 1 = 44/44, 2 = 4/4 + 4/4.

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Puzzle 110

If you have a number of identical wooden cubes and paint each face with either red or green paint, how many cubes can you paint which are distinguishable from each other? No faces can remain unpainted.

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Puzzle 111

Substitute numerical digits for letters in the following expression to make the equation true:

ABCDABCD + ABCD = A x (ABCD) x (ABCD)

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Puzzle 112

When John cashed a check at the bank, the teller interchanged the dollars and cents. After John spent 5 cents, he found that he still had twice the amount of the check. What was the amount of the check?

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Puzzle 113

A toy store keeps white and black marbles in three tin cans. Can #1 is labelled "White", can #2 is labelled "black", and can #3 is labelled "White and Black". The labels, however, have been mixed up so that none of them is correct. If you can remove one marble from each can without looking inside, how many marbles must you remove to enable you to label the cans correctly?

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Puzzle 114

Find numerical digits to substitute for the capital letters in the following expression to make the equation true

(AB) x (AB) x (CDB) = ABABAB

There are three answers

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Puzzle 115

Place the numerals 1 through 8 in the blocks to the left so that no consecutive numbers appear in adjacent squares horizontally, vertically, or diagonally

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Puzzle 116

Andy, Bart, and Chad are boxers. Two are having a fight. (1) The shorter of Andy and Bart is the older of the two fighters. (2) The younger of Bart and Chad is the shorter of the two fighters. (3) The taller of Andy and Chad is the younger of the two fighters. Which of the three is not fighting?

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Puzzle 117

Put + signs in the following expression to make the sum equal 100

1    2     3     4     5     6     7

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Puzzle 118

A farmer has a 20-foot square barn in the middle of a meadow. One end of a 20-foot pole is attached to one corner of the barn with a swivel joint. A goat is tied to the other end of the pole with a 10-foot rope. What is the goat's grazing area?

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Puzzle 119

Find the positive numerical digits for A, B, C, D, E, and F which make the following equations true:

ABC - CBA = DEF

A + B + C = D + E + F = DF

A > B > C

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Puzzle 120

Every day a man's wife meets him at the station at the same time and drives him home. One day he arrives at the station an hour early and begins to walk home along the route his wife always takes. When he meets his wife, she drives him home, arriving 16 minutes earlier than they usually do. How long did the man walk?

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Puzzle 121

A contractor wants to measure the diameter of a circular cross section of a horizontal sewer pipe. All he has to work with is a 48" level, a 36" level, and a 10" rule. He supports each level on the inner surface of the pipe, making sure that each is perpendicular to the axis of the pipe and horizontal. He notes that the distance betweeen the bottoms of the levels is 6 inches. What is the diameter of the pipe?

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Puzzle 122

Find numerical digits to substitute for the capital letters in the following expression to make the equation true. There is only one answer.

ABABAB + 3x(AB) = AB x AB x BAC

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Puzzle 123

A, B, and C are different integers greater than 1. What are the smallest values of these integers that will make the following true?

1. A + B + C = odd integer
2. A x B x C = even integer
3. A / (B + C) = even integer
4. (A + B) / C = even integer

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Puzzle 124

The sum of the ages of a man, his wife, his son, and his daughter is 179 years. Seven years from now the man will be twice as old as his son will be and his wife will be twice as old as his daughter will be. Four years ago the man was three times as old as his daughter was. How old are they now?

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Puzzle 125

In the crossword at the right, replace the letters with numerical digits, such that:

1. a across is Al's age
2. a down is Al's age + Ben's age
3. b across is Al's age + Ben's age + Clark's age
4. Two of Al, Ben, and Clark are the same age; the other is a different age
5. Neither W nor Y is 0

Which two of Al, Ben, and Clark are the same age?

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Puzzle 126

Farmer A said to Farmer B "Give me 17 cows and I will have as many as you." Farmer B replied to Farmer A "Give me 17 cows and I will have twice as many as you." If these are both true statements, how many cows does each farmer have?

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Puzzle 127

In a bag of 300 marbles, 99% are white while the rest are black. How many white marbles must be removed to reduce the percentage of white marbles in the bag to 98%?

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Puzzle 128

Find numerical digits to substitute for the capital letters to make the following equation true:

ABCABC = 7 x (DEF) x (DEF)

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Puzzle 130

In the figure to the right, each letter represents a different digit, 1 through 9. Each of the sums A+B+C, C+D+E, E+F+G, and G+H+I is equal to 13.

What digit does E represent?

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Puzzle 144

Farmer Brown has nine cows. He hires a carpenter to build four enclosures for his cows with the stipulation that each enclosure should hold an odd number of cows. How does the carpenter accomplish this?

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Puzzle 145

In the following figure, replace the letters with numerical digits (0-9) to make a correctly executed addition. Each different letter is to replaced with a different digit and all occurrences of any one letter are to be replaced with the same digit.

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Puzzle 154

In the array of 10 coins shown at the right, there are three rows of four coins each (one horizontal and two vertical).

Move two coins in such a way as to make four rows of four coins each.

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Puzzle 155

John went to the grocery store to buy some fruit. He noticed that 3 grapefruit plus 2 apples cost the same as 4 oranges plus 1 lemon, that 1 grapefruit plus 3 lemons cost the same as 3 oranges plus 1 apple, that 2 grapefruit plus 2 oranges cost the same as 3 apples plus 4 lemons, and that an apple plus a grapefruit plus a lemon plus an orange cost $1.40.

How much did each piece of fruit cost?

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Puzzle 167

A perfectly flexible rope hangs over a frictionless pulley. A chunk of iron is tied to one end of the rope and a monkey of equal weight hangs on the other end. The rope weighs 4 ounces to the foot. The weight of the monkey in pounds is equal to the monkey's mother age in years. The combined age of the monkey and his mother is four years. The monkey's mother is twice as old as the monkey was when the mother was half as old as the monkey will be when he is three times as old as the mother was when she was three times as old as the monkey was. The weight of the rope plus the weight of the iron is half again as much as the difference between the weight of the iron and the weight of the iron plus the weight of the monkey.

How long is the rope?



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Puzzle 207

Find a five digit number in which the first two digits are the same and when read as a two-digit number equals the sum of the second and third digits. The fourth digit is twice the last digit. The sum of all the digits is 16.

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Puzzle 216

John, Paul, and Ray went to the carnival and wanted to try the various rides. When they checked their pockets, they found that they each had a different amount of money. All of John's money was in nickels; all of Paul's money was in dimes; and all of Ray's money was in quarters. They though that it was only fair that they each have the same amount of money to spend so they traded coins between them until they all had the same amount of money. When this happened, they found that each person had given three of his coins to each of the others and that each had $1.70 to spend. How much money did each start out with?

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Puzzle 250

A man and a woman who are not related each have two children. At least one of the woman´s children is a boy. The man´s oldest child is a boy. Which parent has the highest probability of having two boys?

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Puzzle 253

A clock collector noticed that on January 1, 2001, his four grandfather clocks struck 8 AM at exactly the same time. This was unusual because, while one of the clocks kept perfect time, the other three lost time but did so consistently. The second clock lost 2 minutes per day; the third clock lost 2.5 minutes per day; and the fourth clock lost 3 minutes per day.

When will all four clocks next strike 8 AM in unison?

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Puzzle 258

Using each numeral from 1 through 8 just once and only one mathematical sign, write an expression that is exactly equal to 10.

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Puzzle 261

Two spherical cannonballs are made of the same material. The larger cannonball has a 10-inch diameter and weighs 250 pounds. The small cannonball weighs one pound.

What is its diameter?

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Puzzle 307

If 6 apples and 4 grapefruit cost $3.50 and 6 grapefruit and 4 apples cost $4.00, how many grapefruit can you buy for $5.00?

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Puzzle 390

A toy store sells:

Cars for $0.50
Boats for $3.00
Planes for $10.00

If I bought 100 toys for $100, how many of each toy did I buy?

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Puzzle 391

If 3 men can build 5 barns in 7 days

how long will it take 8 men to build 14 barns?

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Puzzle 391

I have an assortment of apples, bananas, cherries and dates. The total number of pieces of fruit is 28. There is at least one piece of each fruit. Each piece of fruit is a whole fruit. Two plus the number of dates minus the number of apples is equal to 2 times the number of bananas. The number of bananas plus the number of cherries is equal to three times the number of apples.

How many of each fruit do I have?

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Puzzle 413

Construct the figure below as follows:

1. Draw horizontal line(a-b), 6 inches long
2. Draw line (b-c) at 30 degrees relative to line (b-a) and 8 inches long
3. Draw vertical line from (a) through (d)
4. Locate (d) on vertical line through (a) such that (a-d) equals (d-c)
5. Using (d) as center, draw a circle through (b)
6. Extend line (b-c) to (e) on the circle

What is the length of (c-e) = x ?

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Puzzle 414

Line AB is 12 inches long
Line AC is drawn 60 degrees from the horizontal
Line BD is drawn 70 degrees from the horizontal
A circle is drawn tangent to AB, AC, and BD
Line CD is drawn parallel to AB and tangent to the circle

How long is Line CD?


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Puzzle 415

A circle is drawn tangent to two sides of a regular pentagon with ten inch sides.

If another circle is drawn centered on the intersection of two other sides of the pentagon as shown and, if the circles are tangent to each other and have the same radius, what is their radius?


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Puzzle 416

If an equilateral triangle with height H is concentric with a circle with radius R, and the area of the triangle outside the circle is equal to the area of the circle outside the triangle, what is the relation between H and R?

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Puzzle 417

A canoeist is travelling upstream in a river. Two miles upstream from his starting point he passes a log floating im the river. He goes upstream for another hour and then turns around and goes downstream. He reaches the starting point at the same time as the log. If the speed of the river current is constant and the speed of the canoe relative to the water is constant, what is the speed of the river current?

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Puzzle 418

A man sold a car for $190, bought it back for $180, then sold it for $200. How much did he make on these transactions?

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Puzzle 419

Using three identical numerical digits and appropriate mathematical operators, write an expression that equals 24. How many different expressions can you find? I found eight.

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Puzzle 420

A missionary had 100 loaves of bread to distribute to a tribe of 100 natives. The native chief decided that each man should get 3 loaves, each woman should get 2 loaves, and each child should get 1/2 loaf. After the distribution was made, as prescribed by the chief, there were no loaves left over. If each man had more than 8 and less than 11 children, how many men, how many women, and how many children are there in the tribe?

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Puzzle 421

A cylindrical barrel is placed in contact with a horizontal floor and a vertical wall and a ladder is placed so that it contacts the floor, the barrel and the wall as shown in the sketch.

If the distance along the ladder from the floor to the contact point with the barrel is 5 feet and the distance along the ladder from the wall to the contact point with the barrel is 10 feet, what is the diameter of the barrel?

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Puzzle 422

A sphere of radius R is to be put into a flat bottomed cylindrical cup of radius R. However, prior to its insertion into the cup, small spheres of radius r are put into the cup. The size and number of small spheres are such that when the large sphere is inserted, it touches each of the small spheres and also the bottom and side of the cup. In addition, each of the small spheres touches the side and bottom of the cup.

What are the relative sizes of the spheres and what is the maximum number of small spheres that can be used in this configuration? (A small hole in the bottom of the cup prevents air pressure problems).

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Puzzle 423

Three salesmen went to a hotel and got a room for $30, paying $10 each. The clerk later noticed that he made a mistake and should have charged them a total of $25. He gave $5 to the bellhop and instructed him to return the money to the salesmen. The bellhop couldn't figure out how to divide the money into three equal parts so he gave each salesmen $1 and kept $2 for himself.

The end result was that the salesmen paid $27 and the bellhop got $2 for a total of $29. What happened to the other dollar?

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Puzzle 424

Two square panes of glass are each hung by a corner so that they overlap as shown. If the distance between the suspension strings is D and the length of the sides of the panes is A , what is the relation between D and A that results in the overlap area of the two panes being the same as the non-overlapped area of each pane.

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Puzzle 425

When using a computer to solve mathematical problems one is often required to evaluate the function:

This function is not available in most computers. Rather the function:

is available from which arcsin(x) can be calculated.

Show that

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Puzzle 426

Two discs of radius R are hung from strings so that they partially overlap each other.

At what value of D do the three areas, A, B, and C, equal each other?

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Puzzle 427

Paul sent Christmas presents to his four brothers Art, Bob, Carl, and Dan. The value of these presents were as follows:

Art's present was equal to Carl's present plus $4

Bob's present was equal to 5/6 of Art's present

Carl's present was equal to 2/3 of Bob's present

Dan's present was equal to 2 times Carl's present

What was the value of all four presents?

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Puzzle 428

You have a 6-minute hourglass and a 10-minute hourglass which you want to use to time the boiling of an egg to 14 minutes

What is the quickest way to accomplish this?

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Puzzle 429

A car rental company has a garage that will house 6 cars less than it owns. If the garage is enlarged by 50%, the garage will house 7 cars more than the rental company owns.

How many cars does the rental company own?

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Puzzle 430

Max and Joe were playing a game of Tic-Tac-Toe (Max marking X and Joe marking O) when the lights went out. Since they couldn't see, they put the game aside and decided to continue the next day. However, by then, they couldn't remember whose move it was. The game board looked like:

The puzzle is to figure out who moves next and wins,

They play by the rules of the game, namely, if on any turn they can win, they must move to win and, if on any turn, they can't win but can block their opponent to prevent a win, they must make the blocking move.

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Puzzle 431

In working with her new balance scale, Jane found the following:

1 fork and 1 cup weighed the same as 3 knives

1 cup weighed as much as 3 spoons and 1 knife

3 forks weighed the same as 1 knife and 1 spoon

How many spoons weigh the same as 1 cup?

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Puzzle 432

Using all digits 1-9 just once, but no zeros, write a fraction that equals precisely 1/9.

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Puzzle 433

IF (4 + 5 + 6) ^ (33) = A

and (6 - 5 - 4) ^ (33) = B

What is the value of (4 + 5 + 6) ^ (A + B)?

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Puzzle 434

An auto traveled 10 laps around the Indy Motor Speedway at an average speed of 50 miles per hour.

How fast would the auto have to travel for the next 10 laps to have an average speed of 100 miles per hour for the entire 20 laps?

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Puzzle 435

What combination of 9 US coins can be used to make up any value from 1 cent to 99 cents inclusive?

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Puzzle 436

Using all digits 1-9 just once, but no zeros, write a fraction that equals precisely 1/7.

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Puzzle 437

What is the largest amount on money that you can have using $1, $2, $5, $10, $20, and $50 bills without having to make exact change for a $100 bill?

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Answer to Puzzle 109

There are more than one way to express many of the cardinal numbers using four 4's. Following is one solution:

1=44/44	7=44/4-4	13=(44/4)+√4	19=4!-4-(4/4)	25=4!+(√4x√4/4)
2=(4/4)+(4/4)	8=4+4+4-4	14=4+4+4+√4	20=44/(√4+√4)	26=(44/√4)+4
3=(4+4+4)/4	9=(4x√4)+4/4	15=(44/4)+4	21=4!-4+(4/4)	28=4!+4+4-4
4=4+√4-(4/√4)	10=(44-4)/4	16=4+4+4+4	22=(44/4)x√4	28=4!+4+4-4
5=√4+√4+(4/4)	11=44/(√4+√4)	17=(4x4)+4/4	23=4!-(√4x√4)/4	29=4!+4+(4/4)
6=4+4-(4/√4)	12=(44+4)/4	18=(4x4)+4-√4	24=(44/√4)+√4	30=4!+4+(4/√4)

Answer to Puzzle 110

How many cubes can you paint the faces with red or green paint that are distinguishable from each other?

There are 10 different cube patterns: (1)-All red; (2)-All green; (3)-1 red face; (4)-1 green face; (5)-2 red faces adjacent; (6)-2 green faces adjacent; (7)-2 red faces opposite; (8)-2 green faces opposite; (9)-2 red faces opposite and one red face connecting the other two (green faces have the same pattern); (10)-3 red faces meeting on a corner (green faces have the same pattern).

Answer to Puzzle 111

Solving for:

ABCDABCD + ABCD = A x (ABCD) x (ABCD)

A=3, B=3, C=3, D=4

33343334 + 3334 = 3 x 3334 x 3334

Answer to Puzzle 112

Let the original check be for A dollars and B cents or (100A + B) cents.
John actually received B dollars and A cents or (100B + A) cents
After spending 5 cents, John had (100B + A -5) cents
Then, (100B + A -5) = 2 x (100A + B) = (200A + 2B)

Reducing, 98B = 199A + 5

To solve this equation, let B = 2A + z

Substituting, this gives: 196A + 98z = 199A + 5 or 98z = 3A + 5

By inspection, a solution is: z = 1 and A = 31

Then B = 63

Thus, the original check was for $31.63

Answer to Puzzle 113

You need to remove only one marble from can #3 labelled "White and Black". If you withdraw a white marble, you know that can #3 should be labelled "White" since you know that the "White and Black" label is incorrect.
Can #2 should then be labelled "White and Black" since it was originally labelled "Black" which we know is wrong, and it can't be labelled "White" since can #3 is "White".
Then, can #1 should be labelled "Black"
Similar logic applies if the marble withdrawn is black.

Answer to Puzzle 114

1) A=2, B=1, C=4, D=8 -- 21 x 21 x 481 = 212121
2) a=3, B=9, C=2, D=5 -- 39 x 39 x 259 = 393939
3) A=9, B=1, C=1, D=1 -- 91 x 91 x 111 = 919191

Answer to Puzzle 115

Answer to Puzzle 116

Chad is not fighting.

Answer to Puzzle 117

The answer is: 1 + 23 + 4 + 5 +67

Answer to Puzzle 118

The goat's grazing area is 2042 square feet.

Answer to Puzzle 119

When A > C and ABC - CBA = DEF, then E = 9 and D + F = 9 (Verify this using any numbers)

Then A + B + C = D + E + F = 18 = DF, thus D = 1, F = 8 and DEF = 198

There are seven combinations of A, B, and C for which A > B > C and A + B + C = 18

Of these, only A = 7, B = 6, and C = 5 results in ABC - CBA = 198

Therefore, A = 7, B = 6, C = 5, D = 1, E = 9, and F = 8

Answer to Puzzle 120

If they arrived home 16 minutes earlier than usual, the wife's trips from and to their home were each 8 minutes shorter than usual. Thus, she met him 8 minutes earlier than usual. By this time, the man had walked 52 minutes.

Answer to Puzzle 121

The pipe has an internal diameter of 60 inches.

Answer to Puzzle 122

Solving for: ABABAB + 3x(AB) = AB x AB x BAC:

A = 2, B = 4, C = 1

So that 242424 + 3 x 24 = 24 x 24 x 421

Answer to Puzzle 123

A, B, and C are different integers greater than 1. What are the smallest values of these integers that will make the following true?

(1) A + B + C = odd integer
(2) A x B x C = even integer
(3) A / (B + C) = even integer
(4) (A + B) / C = even integer

From Equation (2), there must be at least one even integer.
From Equation (1), there must be another even integer and one odd integer
From Equation (3), A ≥ 2 (B + C); A is an even integer.
From Equation (4), B must be an even integer.

Thus, C is an odd integer.

The smallest values of B and C are 2 and 3.

From Equation (3), the smallest value of A is 10.

Answer to Puzzle 124

The man is 67; the wife is 57; the son is 30; and the daughter is 25.

Answer to Puzzle 125

WX is Al's age and W > 0
       Thus, Al's age is 10 or greater
WZ is Al's age + Ben's age;
       Thus, Ben's age is WZ-WX, a single digit; B=Z-X
YZ is Al's age + Ben's age + Clark's age;
       Thus, Clark's age is YZ-WZ, a multiple of 10; (Y-W) 0
Ben is less than 10 and the other two are 10 or older;
       Thus, Al and Clark are the same age; X=0; W=Y-W; y=2W
In the 36 possible solutions:
       W = 1, 2, 3, or 4
       X = 0
       Y = 2 x W = 2, 4, 6, or 8
       Z = 1, 2, 3, 4, 5, 6, 7, 8, or 9

Answer to Puzzle 126

The statements of the puzzle are:

1) A + 17 = B - 17
2) B + 17 = 2 (A-17) = 2A - 34
       Rearrange 2)
3) 2A - 34 = B + 17
       Subtract 1) from 3)
4) A - 51 = 34
5) A = 34 + 51 = 85
       Substitute in 1)
6) 85 + 17 = B - 17
7) B = 85 + 17 + 17 = 119

Answer to Puzzle 127

There are three black marbles in the bag. Originally they constituted 1%. To increase their percentage to 2%, the total number must be halved. Thus 150 white marbles must be removed.

Answer to Puzzle 128

Find numerical digits to substitute for the capital letters to make the following equation true:

ABCABC = 7 x (DEF) x (DEF)

ABC = 143 = DEF thus 143143 = 7 x 143 x 143

Answer to Puzzle 130

E is equal to 4. Here is one possible solution.

Answer to Puzzle 144

Answer to Puzzle 145

Here is one solution to the puzzle. There are 23 other solutions.

Answer to Puzzle 154

Move the red coins to the blue positions as shown.

Answer to Puzzle 155

Apples cost 20 cents; grapefruit cost 50 cents; lemons cost 30 cents; and oranges cost 40 cents.

Answer to Puzzle 167

Rewrite the puzzle, inserting for reference, times a, b, c and the present as follows:

A perfectly flexible rope hangs over a frictionless pulley. A chunk of iron is tied to one end of the rope and a monkey of equal weight hangs on the other end. The rope weighs 4 ounces to the foot. The weight of the monkey in pounds is equal to the monkey's mother age in years. The combined age of the monkey and his mother is four years. The monkey's mother (at present) is twice as old as the monkey was (time c) when the mother was half as old as the monkey will be (time b) when he is three times as old as the mother was (time a) when she was three times as old as the monkey was. The weight of the rope plus the weight of the iron is half again as much as the difference between the weight of the iron and the weight of the iron plus the weight of the monkey.

How long is the rope?

Let W = weight of monkey in pounds
    Am = age of monkey in years at present
    Amm = age of monkey's mother in years at present
        K = difference in ages = Amm - Am


From the puzzle:
    Amm + Am = 4
    W = Amm
        Weight of iron = weight of monkey = W

At time a, L years ago, when mother was 3 times as old as the monkey:
    Monkey's age = Am - L
    (*)Mother's age = Amm - L = 3 (Am-L) = 3Am - 3L
        But Am = 4 - Amm (from the puzzle)
        Substituting: Mother's age = Amm - L = 3(4-Amm) - 3L = 12 - 3Amm - 3L
            Rearranging: 2L = 12 - 4Amm
                (****)L = 6 - 2Amm

At time b, when monkey was 3x as old as mother was at time a:
    Mother's age = 3Am - 3L from (*)
        (**)Monkey's age = 3 (3Am - 3L) = 9Am - 9L = 9 (Am - L)

At time c, when mother was half as old as monkey at time b:
    Mother's age = 4.5 ( Am-L) from (**)
        (***)Monkey's age = Mother's age - K = 4.5 (Am - L) - K

At the present time:
    Mother's age = 2 times monkey's age at time c
        = 9 (Am - L) - 2K from (***)
    But K = Amm - Am (by definition)
        Therefore Mother's age = Amm = 9Am - 9L - 2Amm + 2Am
    Consolidating: Amm = 11Am - 9L - 2Amm
            3Amm = 11Am - 9L
    Substituting from (****): 3Amm = 11Am - 9(6 - 2Amm) = 11Am - 54 + 18Amm
        Rearranging: 54 = 11Am +18Amm - 3Amm = 11Am + 15Amm
            Substituting: 54 = 11(4 - Amm) + 15Amm = 4Amm + 44
            Rearranging: 10 = 4Amm
    Therefore: Mother's age = Amm = 2.5
        Monkey's age = 4 - Amm = 1.5

Weight of iron in pounds = Monkey's age in years (from puzzle)
    Therefore W = 2.5
        Weight of monkey = Weight of iron = W (from puzzle)
Let weight of rope = Wr
    Weight of iron + weight of monkey = 2W
        Then Wr + W = 1.5 (2W - W) = 1.5 W (from puzzle)
            Rearranging: Wr = 0.5W = 0.5(2.5) = 1.25 pounds = 20 ounces

Therefore length of rope = 20 ounces / 4 ounces/foot = 5 feet

Answer to Puzzle 207

Since the first two digits when read as a two-digit number equal the sum of the other two digits, this two-digit number must be less than 19. Since the first two digits are the same, they must both be 1 and the two-digit number is 11. Thus, the sum of the first four digits is 13 (first digit = 1; second digit = 1, sum of second and third digits = 11). Because the total of all the digits is 16, the fifth digit must be a 3. The fourth digit is twice the last digit, so it must be 6. The third digit is then 5 (13 - 1 - 1 - 6).

Thus, the initial number is 11563.

Answer to Puzzle 216

After they had traded coins:

John had 0.25 * 3 + 0.10 * 3 + X * 0.05 = 1.05 + X * 0.05 = $1.70

Paul had 0.25 * 3 + 0.05 * 3 + Y * 0.10 = 0.90 + Y * 0.10 = $1.70

Ray had 0.10 * 3 + 0.05 * 3 + Z * 0.25 = 0.45 + Z * 0.25 = $1.70

Rearranging and solving each equation:

X = (1.70 - 1.05) / 0.05 = 13

Y = (1.7- - 0.90) / 0.10 = 8

Z = (1.70 - 0.45) / 0.25 = 5

Thus, at the start:

John had (13 + 6) * 0.05 = $0.95

Paul had (8 + 6) * 0.10 = $1.40

Ray had (5 + 6) * 0.25 = $2.75

Answer to Puzzle 250

When there are two children in the family, there are four possible distributions, each equally likely:

	1	2	3	4
Older:	boy	boy	girl	girl
Younger:	boy	girl	boy	girl

The woman is in Group 1, 2, or 3. But in only one of these groups are there two boys. Therefore her chances of having two boys is 1 in 3.

The man is in Group 1 or 2 so his chances of having two boys is 1 in 2.

Answer to Puzzle 253

The answer is September 2, 2009.

There are 1440 minutes in a day. To strike in unison, the clocks must each lose a multiple of 1440 minutes. Clock 2 loses 1440 minutes in 720 days; Clock 3 loses 1440 minutes in 576 days; and Clock 4 loses 1440 minutes in 480 days. The least common multiple of 720, 576, and 480 is 2880. Thus, they will all strike together in 2880 days. 2880 days after 8 AM January 1, 2002 is 8 AM September 2, 2009

Answer to Puzzle 258

The answer is:

4 (1578 / 263) = 10 (parentheses added for clarity)

Answer to Puzzle 261

If the weight ratio is 250:1, then the volume ratio is also 250:1. So the diameter ratio is:

Thus, the diameter of the small cannon ball is 10 / 6.3 = 1.59 inches.

Answer to Puzzle 307

The equations involved are:

(a) 6A + 4G = 3.5 ;
(b) 4A + 6G = 4.0;
(c) Multiply (b) by 1.5 → 6A + 9G = 6.0;
(d) Subtract (a) from (c) →5G = 2.5;
(e) Therefore, G = 0.5;

10 grapefruit = 10G = $5.00

Answer to Puzzle 390

There are two answers

80 Cars     =$40          94 Cars     =$47
20 Boats = $60           1 Boat       = $3
100         $100            5 Planes    = $50
                                     100             $100

Answer to Puzzle 391

Eight men can build barns in 3/8 the time that 3 men could build the same barns.

Fourteen barns can be built in 14/5 the time that 5 barns can be built.

Thus, 8 men could build 14 barns in 7 x 3/8 x 14/5 = 7.35 days.

Answer to Puzzle 392

The equations to be solved are:

a) A + B + C + D = 28
b) 2 + D - A = 2B
c) B + C = 3A

These equations lead to:

d) D + 4A = 28      from a) and c)
e) D = 28 - 4A
f) D = A + 2B - 2      from b)
g) A + 2B - 2 = 28 - 4A      from e) and f)
h) 2B + 5A = 30
i) B = (30 - 5A)/2
j) C = 3A - B      from c)

Make a table for various values of A:

A	B	C	D
1	12.5	-9.5	24
2	10	-4	20
3	7.5	1.5	16
4	5	7	12
5	2.5	12.5	8
6	0	18	4
7	-2.5	23.5	0

A = 4 is the only case for which A, B, C and D are non-zero positive integers, thus these values for A, B, C and D are the answers to the puzzle.

Answer to Puzzle 413

This puzzle is most easily solved by noting that when two points are equally distant from the center of a circle, as are a and c, and each point divides a chord of the circle into two segments, the product of the two segments produced by one point is equal to the product of the two segments produced by the other point.

Extend line (b-a) to (f) on the circle

Since (d) is the center of the circle,     (a-f)=(a-b)=6

Thus 8y = 6x6 = 36

y = 4.5 inches

Answer to Puzzle 414

Construct radius R intersecting Line AB at 90 degrees

Then, a + b = 12

R = a tan 30 = b tan 35

a tan 30 = (12-a) tan 35

0.577 a = 0.700 * 12 - 0.700 * a

1.277 a = 8.4         a = 6.578 in

b = 12 - a = 5.422 in

R = 6.578 tan 30 = 3.796 in

h = 2R = 7.592 in

e = 7.592*tan 30

f = 7.592*tan 20

CD = 12 - e - f

CD = 12 - 4.383 - 2.763

CD = 4.854 in

Answer to Puzzle 415

Side A = 2R

Side B = 10 * sin(72) - R = 9.511 - R

Side C - 10 + 10 * sin(18) - R * tan(36) = 13.09 - .727 * R


(Side A)² = (Side B)² + (Side C)²

4R² = (9.511-R)² + (13.09-.727*R)²

2.471R² + 38.055R - 261.807 = 0

R = -38.06 ± √ (1448.2 + 4*2.471*261.8) / 4.942

R = 5.155

Answer to Puzzle 416

AT = Area of triangle = H*(H/cos30)/2 = H2/1.732 = 0.577 H2


AC = Area of circle = πR²

CA = area in common between triangle and circle
            COT = area of circle outside triangle = AC - CA
                        TOC = area of triangle outside circle = AT - CA

When COT = TOC ; AC - CA = AT - CA and AC = AT
           Thus 0.577 H² = πR²
                And H/R = √( 3.1416/.577) = 2.333

Answer to Puzzle 417

Let r = speed of river in mph.

Let c = speed of canoe in still water in mph.

After passing the log going upstream, the canoeist goes upstream for an hour at a speed of (c-r)mph

        for a distance of (c-r)miles.

He then goes downstream a distance of (c-r)miles at a speed of (c+r)mph in ((c-r)/(c+r)) hr.

He then goes downstream for 2 miles at a speed of (c+r) in (2/(c+r))hr.

The time required for the canoeist after passing the log going upstream and back to the starting point

        is (1 +(c-r)/(c+r)+ 2/(c+r)) hours.

After the canoe passes the log going upstream, the log floats downstream 2 mi to the starting point

        at a speed of r mph in (2/r) hours.

Then, 1 + ((c-r)/(c+r)) +(2/(c+r)) = 2/r

        ((c+r) + (c-r) +2)/(c+r) = 2/r

                2c + 2 = 2(c+r)/r

                2cr + 2r = 2c +2r

                cr = c

                r = 1 mph

Answer to Puzzle 418

The man started out with a car worth $190. He ended up with $210 in his pocket. He made $20

Answer to Puzzle 419

The expressions that I found are:

• 8 + 8 + 8 = 24
• 22 + 2 = 24
• 3³ - 3 = 24
• 4 * (4 + √4) = 24
• (3 + 3/3)! = 24
• ((4/4) * 4)! = 24
• (2 * √2 * √2)! = 24
• (2 + (√2 * √2)! = 24

Answer to Puzzle 420

The equations to be solved are:

M + W + C = 100
3M + 2W + C/2 = 100
C > 8M and C < 11M

Consider the first 2 equations:

M + W + C = 100
6M + 4W + C = 200

Subtract the first from the second:

5M + 3W = 100
W = (100 - 5M)/3

The only values of M that give integer values of W are:

M=2 : W=30 : C=68
M=5 : W=25 : C=70
M=8 : W=20 : C=72
M=11 : W=15 : C=74
M=14 : W=10 : C=76
M=17 : W=5 : C= 78
M=20 : w=0 : C=80

The only values of M for which C is greater than 8M are M = 2, 5 or 8.
The only values of M for which C is less than 11M are M = 8 and greater.
Thus, M=8 is the only value that satisfies the puzzle.

Therefore, there are 8 men, 20 women and 72 children in the tribe.

Answer to Puzzle 421

Make additions to the sketch as shown.

Tan(b) = R/5
Tan(2b) = (10 + R)/(5 + R)
Tan(2b) =(2 tan(b))/(1 - tan^2(b)),from math tables

(10+R)/(5+R)=(2R/5)/(1-(R^2)/25)

=10R/(25-R^2)
=10R/(5-R)(5+R)
10+R =10R/(5-R)
(10+R)(5-R)=10R

50+5R-10R-R^2 = 10R
50-15R-R^2 = 0
R^2 + 15R - 50 = 0
R = (-15 ± ?(225 + 200))/2
R = (-15+20.616)/2 = 5.616/2 = 2.808 ft
D = 5.616 ft

Answer to Puzzle 422

Side view of cup and spheres

R + r + 1.414r = 1.414R
2.414 r = 0.414 R
r = 0.414 R / 2.414 = .171 R



Bottom view of cup and small spheres

Θ = sin-1(.171R/(R-.171R))= sin-1 (.206)=11.888deg
2Θ = 23.776 deg subtended by each small sphere at center of cup
360/23.776 = 15.141 small spheres around cup

15 uncut small spheres around cup

Answer to Puzzle 423

There isn't any problem. The salesmen paid a total of $27 of which the clerk got $25 and the bellhop got $2.

The confusion is in the asking of the question.

Answer to Puzzle 424

Add dimensions S and M to the drawing as shown.

Area overlapped by both panes (light blue) = S²
Area not overlapped in one pane (red) = A² - S²

But A² - S² = S² (from problem statement)

Thus, A² = 2 x S²
A = √2 x S
S = A / √2
M = √2 x S

From drawing:

D = √2 - √2 x S
D = √2 x A - A
D = 0.414 x A

Answer to Puzzle 425

Answer to Puzzle 426

Given that area A = area B = area C,

Divide area B into two halves, B/2, and add dimension X = 1/2 D

The area B/2 can be calculated using the following formula from handbook tables:

When arctan is used in place of arcsin to permit use of a computer, this equation becomes:

The conditions of the puzzle are that:

At this point it is noted from the sketch that, if R is changed by a factor m, X is also changed by the factor m and all the areas are changed by the factor m². Areas that were equal to one another before the change in R will still be equal after the change in R and the relation between R and X will not change with changes in m. Thus, we can calculate the value of X/R for any value of R, and especially R=1, which simplifies the calculation significantly.

Now substitute different values of X in the above expression until the solution for π/4 = 0.78539

With a computer it may be found that setting X = 0.40397 gives a value for π/4 of 0.78539. Thus, the relation between X and R is X = 0.40397 R

Therefore, D = 2X = 0.80794R

Answer to Puzzle 427

From the statement of the problem:

A = C + 4
B = 5/6 A
C = 2/3 B
D = 2 C

Therefore:

C = (2/3) * (5/6) * A = 5/9 A
A = C + 4 = 5/9 A + 4;      4/9 A = 4;      A = 9
C = 5/9 A = 5
B = 5/6 A = (5/6) * 9 = 7.5
D = 2 * C = 10

Therefore: the total of all these gifts is $31.50

Answer to Puzzle 428

Follow these steps:

Bring the water to a boil

Start both the hourglasses at zero. Put the egg in the water.

At time 6, the 6-minute hourglass runs out. Turn it over to start it again

At time 10, the 10-minute hourglass runs out. The egg will have boiled 10 minutes. The 6-minute hourglass has been running 4 minutes. Turn the 10-minute hourglass over to start it over again.

At time 12, the 6-minute hourglass runs out for the second time. The egg has been boiling 12 minutes. The 10-minute hourglass has been running 2 minutes. Turn the 10-minute hourglass over so it will have 2 minutes to run.

At time 14, the 10-minute hourglass runs out again. Remove the egg from the boiling water. It will have boiled for 14 minutes.

Our thanks to Nick Dare for a corrected answer to this problem.

Answer to Puzzle 429

A car rental company has a garage that will house 6 cars less than it owns. If the garage is enlarged by 50%, the garage will house 7 cars more than the rental company owns.

Let G = number of cars that will fit in the garage and N = number of cars owned.

then 1.5 G = N + 7
G = N - 6
0.5 G = 13
G = 26 and N = 32

Answer to Puzzle 430

Identify the boxes in the puzzles as shown:


Consider whether O could have been put into vacant box a on the last move. No, according to the rules, it would have to go into box e.

Could O have gone into vacant box d on the last move? No, it would have to have gone into box e.

Could O have gone into vacant box f on the last move? No, it would have to have gone into box e.

So, Joe did not move last.

Could X have gone into box b on the last move? No, it would have to go into box e.

Could X have gone into box c on the last move? No, it would have to go into box e.

Could X have gone into box g on the last move? Yes, it could have. It violates no rules

Thus, Max had the last move. Now it's Joe's turn and he wins by putting O in box e.

Answer to Puzzle 431

Setting F for Fork, C for Cup, and K for Knife, the equations to be solved are:

a.     F + C = 3K
b.     C = 3S + K
c.      3F = 1K +S

And we want to calculate X = C/S

d.     3K - F = 3S + K     from a. and b.
e.     -F = 3S - 2K
f.     F = 2K - 3S
g.     3F = 6K - 9S
h.     6K - 9S = K + S     from g. and c.
i.     5K = 10S
j.     K = 2S
k.     C = 5S     from b. and j.

Therefore, C/S = 5

Answer to Puzzle 432

6381 / 57429

Invert that to see that 57429 / 6381 = 9 exactly

Answer to Puzzle 433

A = (4 + 5 + 6)^(33) = 3^(33) = 5.56 x 10^15

B = (6 - 5 - 4)^(33) = -3^(33) = -5.56 x 10^15

A + B = 0

(4 + 5 + 6)^0 = (15)^0 = 1

Answer to Puzzle 434

10 laps = 25 miles. At 50 mph, this would take 30 minutes.

20 laps = 50 miles. At 100 mph, this would take 30 minutes.

After 10 laps at 50 mph, the entire times is used up. So it would be impossible to go fast enough during the second 10 laps to average 100 mph.

Answer to Puzzle 435

The coins required are: $.50; $.25; $.10; $.05; $.05; $.01; $.01; $.01; $.01 (1 half-dollar, 1 quarter, 1 dime, 2 nickels, 4 pennies).

Answer to Puzzle 436

2394 / 16758 = 1 /7

Invert to see that 16758 / 2394 = 7 exactly

Answer to Puzzle 437

The largest amount that you can have is $143 made up of 1 $50 bill, 4 $20 bills, 1 $5 bill and 4 $2 bills.