Three Squares in 2008

Here's another request from a reader:

"How can I write 2008 as the sum of three squares?"

There are four ways you can write 2008 as the sum of three squares, can you find all of them?

Simple Algebra

A reader of this page asks:

"If x²+y² = 36 and (x+y)² = 64, what is the value of x∙y?"

Help a fellow puzzle fan discover the values of x and y.

7 and 7 and 7 and 7 is 56

Using four 7s and any of the basic arithmetic operators (+, -, x, ÷) can you make 56?


7    7    7    7 = 56

Taking Notes

If six boys can fill up six notebooks in six weeks and four girls can fill up four notebooks in four weeks, how many notebooks can a class of twelve boys and twelve girls fill up in twelve weeks?

Change for a Dollar?

In the American money system, there are five coins in regular use. The penny is worth 1 cent, the nickel is worth 5 cents, the dime is worth 10 cents, the quarter is worth 25 cents, and the half-dollar is worth 50 cents.

If you use no more than 4 of any type of coin, how many different ways can you make change for 1 dollar (100 cents)?

More Barrels and More Pellets

In the puzzle Barrels Full of Pellets you were asked to find a way to discover which single barrel contained pellets that were slightly heavier than all the other barrels. This brain teaser is based on that one, but is considerably more difficult:


You are presented with ten barrels of pellets, but this time you are told that some or all *may* have 2 gram pellets and the rest have 1 gram pellets. You are given a scale for measuring and you are only allowed to take a measurement one time.

How do you find out which of the ten barrels have 1 gram pellets and which of the barrels have 2 gram pellets?

Hobos and Cigars

A certain hobo can make cigars from discarded cigar butts. He finds that with 5 cigar butts he can make one whole cigar.

One month he collects 25 butts. How many cigars can he make?

Artful Arithmetic

Professor Egghead had a student who was not very good with fractions and thought she had stumbled upon a quick way of discovering which of two fractions was the larger.

When she was asked to find the larger between 2/5 and 3/7 she simply subtracted the numerator from the denominator in each fraction, replacing them with 2/3 (2/(5-2)) and 3/4 (3/(7-3)) respectively, which she then replaced with 2/1 and 3/1, using the same method, and concluded that the first, 2/5, was the smaller.

Professor Egghead was impressed with her method. Was her method valid or was it complete nonsense and her correct answer only a lucky coincidence.

5 Times 2 = 7?

Using only basic arithmetic operations make 7 out of five '2's.


2    2    2    2    2 = 7


You can use '+', '-', 'x', or '÷' between the '2's.

Odd Arithmetic

Find four consecutive odd numbers that add up to 80.

Find five consecutive odd numbers that add up to 85.

The Motel Room

Three businessmen are in Cleveland for a convention. Since they are on a budget, they decide to share a room at a motel that charges $30 per night ($10 per man). The motel manager is in a good mood that night and decides to reimburse some of the money. He gives the bell boy $5 and tells him to give it to the three men. However, the bell boy is dishonest and figures that you cannot divide $5 evenly among three men, so he gives back $1 to each man and keeps the other $2 for himself.

Now, the businessmen have each paid $9 for the room, or $27 all together, and the bell boy has $2, for a total of $29.

What happened to the other $1?

A Binary Crossnumber Puzzle

This puzzle consists completely of binary numbers, so all the characters needed to fill in the squares will be 0s or 1s. The crossword is a 4x4 square grid, so all numbers will be written in binary, with 4 digits; e.g., 1 will be 0001, 2 will be 0010, and 4, 0100. The NOT operation changes all 0s to 1's and all 1s to 0s; e.g., NOT(0110) is 1001 and NOT(1010) is 0101.

Rows (Across): 1. "2 Down" x 2 2. A triangular number 3. The cube of ("4 Down" - 2) 4. "3 Across" + "3 Down"

Columns (Down): 1. NOT "2 Across" 2. NOT "1 Across" 3. "2 Across" x 2 4. "4 Across" - "1 Across"

Needle in a Haystack

How many needles can you find in the haystack below? The word 'needle' may appear horizontally, vertically, or diagonally in any direction, but all six letters will appear in a straight line.


n e d e d
d e d e d l e l l
e e e e n n e e l e e
n n d e d l e d e e d n l
e l e e e e d e n n l d e
e e l n l e e e e n e d e d n
l l n n e e e e n d n e e e e
e e d e e l d e l e d d n d l
e e d e e l e d l n l l e e e
e n l e e e e e e d e d n l l
n n e e n n e n e e e e e e d
e e l d e e d d n n l e e l e
e d d l n l e l e e l n n e e


The Magic Keyring

Engrave numbers on 5 keys on a circular keyring so that the numbers on adjacent groups of keys sum to any value between 1 and 21 inclusively.

For example, 1,1,3,6,6 can sum up to any number between 1 and 17 (1=1, 1+1=2, 3=3, 3+1=4, 3+1+1=5, 6=6, 6+1=7, 6+1+1=8, 6+3=9, 6+3+1=10, etc).

Bobbing for Apples

Professor Egghead's secretary, Mrs. Canton, wanted to buy all the grocer's apples for a church picnic. When she asked how many apples the store had, the grocer replied, "If you add 1/4, 1/5, and 1/6 of them, that would make 37."

How many apples were in the store?

The Red Herring

A 'Red Herring' is a plot device used in literature to distract the reader away from the main event of the story by focusing on a minor event or describing characters in ways that go against our sense of the way those character should be. In cryptography, a red herring is a second hidden message that is intended to be discovered more easily so that the real message remains hidden to anyone who might intercept the transmission and break the red herring code. Only the intended receiver would know the key to unlocking the real message.

The cryptogram below, with two hidden messages, is a prime example of a red herring. One message is fairly easy to decipher, especially if you were able to decode an earlier puzzle that appeared here: http://puzzlepage.blogspot.com/2008/01/find-hidden-message.html. The second message, the one that's the real message, is hidden using a different code that has been made to fit in the same grouping of numbers. This is an extremely difficult cryptogram to solve, so feel free to ask for hints in the comment section.


21941648698194164869819416486981
54961847952716486981947648697358
39114467658829115524463869851941
76487962174268859915413638294575
51947682873991174467835921746687
82992113426384971634855658399727
12432613829431624856389791172446
83953124636885997711344766849911
44758746436849618496184961849618
49361849618898184961849618496184
69819416486981941648698194164869
81961635248698194164869819416486
89915214466889912144668899114466
88279911446688995114466889911446
75879618496188921246648691144666
89921347658591134764869871924164
86921354456289291314419885991234
61839518465768533281559123134362
84931546687899361547678297124312
44951746678897194362778135951543
64856618399613949711429889811444
48896919466819961882828694114914
49981941698618994964219181649644


Good luck!

I Want Candy!

Donna bought one pound of jellybeans and two pounds of chocolate for $2. A week later, she bought four pounds of caramels and one pound of jellybeans, paying $3. The next week, she bought three pounds of licorice, one pound of jellybeans, and one pound of caramels for $1.50.

How much would she have to pay on her next trip if she bought a pound of each of the four kinds of candy?

12 I in a F

Another reader of this page asks:

"I've been given a puzzle where I have to find words that fit in place of the single capital letters in this sentence:


There are 12 I in a F.

The single letters are to be replaced with words that start with the provided letter."


I'm sure most of you can figure it out soon enough--hopefully the reader who asked the question won't stub their toe in their haste to solve this little stumper.

Pigs in a Pen

Can you put nine pigs in four pens so that there are an odd number of pigs in each pen?

4 + 5 = 10?

In the figure below are four straight lines. Can you add five straight lines to the figure to end up with ten?

It's Hip to be Square

Arrange two of each of the digits 0 through 9 to form a 20 digit number. The number may not begin with 0. Then score the number as follows:

For every two consecutive digits that form a perfect square, score two points. For every three consecutive digits that that form a perfect square, score three points. A four digit square scores four points, and so on.

For example, if your number was 58738219024719503664, you would get two points for 49, two points for 36, two points for 64, and six points for 219024 for a total of 12 points. You may not count 036 as a three digit square.

What is the maximum number you can score?

5 Point Star

Fill in the ten circles with numbers so that the sums of numbers along each of the five lines are the same value. The numbers do not need to be contiguous. What is the smallest sum you can make using whole numbers? Can you find a solution with sums less than 41?



Click on the picture to view a larger version of the image.

United States Geography

1. What is the largest lake completely inside the US border?

2. Which city is further west, Pittsburgh or Miami?

3. Are there any states that share borders with eight other states?

4. Are there any states that have borders made up of four straight lines?

5. Is there any part of the continental United States that lies north of the 49th parallel?

6. Which of the 50 states is furthest south?

7. Which of the 50 states is furthest west?

8. If you draw a north/south line through the center of Savannah, Georgia, through which countries in South America would that line pass?

9. If you draw the shortest line from Washington, D.C. to Tokyo, Japan, which countries does that line pass through?

10. Which state has the longest coast line?

Two and One-Half Boys

A colleague of Professor Egghead was explaining to him about his childhood. He said that he was one of five children and that half of them were boys.

How can this be true?

About Two Pounds - A Very Difficult Cryptogram

The following text has a message hidden in it.

Was; I not done, chopping, back some brush with my mitten; when I heard an angel say "You're; not going to see; much here."

"Why won't, I see much?" I, said. "I; want to see something now!"

"My; aren't we in a rush?"; she, replied.

"Shouldn't; I be? It's my way;", said I.

"Oh yes; it is. I forgot you were Mr, X" she said.

Do you think the angel flew away with, X?

The Victorian Boat Ride

There are three couples trekking across the countryside when they come to a river that is too deep and to wide for them to swim across. On the near shore is a boat which they can use to cross the river, but there are some complications that keep them from crossing together.

First of all, the boat is only large enough to carry two people at a time.

Secondly, the couples, being Victorian, are very mindful of proper etiquette and it would be completely improper for a woman to be in the company of another man unless her husband is present.

How can all six people cross the river with a maximum of 2 people crossing at any one time and no woman being in the presence of a man unless her husband is also present?

Star Spangled Triangles

There are ten unique triangles in the figure below. The points, A, B, D, E, and F are obvious. You can also make triangles using opposing points along with the center pentagon; the combinations ACE, ACF, BCD, BCF, and DCE also are triangles.

Can you draw one line through the star to end up with more than 15 unique triangles?

Remove the Tokens

Place tokens in the nine center squares of a 5 x 5 grid. Remove the tokens one at a time by performing jumps, like in the game of checkers. You may jump in any direction, but you may only jump one square at a time. Can you remove eight tokens with the remaining token positioned in the center square?

Gettin' Hitched

This riddle comes from a book entitled "The American Tutor's Assistant", dated 1791.

When first the marriage knot was ty’d
Between my wife and me,
My age was to that of my bride
As three times three to three
But now when ten and half ten years,
We man and wife have been,
Her age to mine exactly bears,
As eight is to sixteen;
Now tell, I pray, from what I’ve said,
What were our ages when we wed?

Cryptic Simple Math

What is the answer to the question hidden in the code below?

17 17 04 31 31 14 04 00 04 14
17 17 10 04 04 17 28 04 28 17
17 17 10 04 04 16 04 04 04 17
21 31 17 04 04 14 04 31 04 02
21 17 31 04 04 01 04 04 04 04
10 17 17 04 04 17 04 04 04 00
10 17 17 04 31 14 31 00 31 04

The Lawn Boys

If a boy and a half can mow a lawn and a half in a day and a half, how many and a half can mow a dozen and a half in a week and a half?

M & M

What comes next in the following sequence?

1212210202001, 224000000, 11333311, 623351, 290221, ...?

Who Gets Paid the Most?

Albert's weekly paycheck amounts to $250 plus 2/5 his weekly paycheck.

Charles' weekly paycheck amounts to $350 plus 3/5 his weekly paycheck.

Jane's weekly paycheck amounts to $450 plus 4/5 her weekly paycheck.

Susan's weekly paycheck amounts to $150 plus 1/5 her weekly paycheck.


How much does each person make in a week?

2 x 2 = 3

Replace the letters in the crypt-arithmetic problem below with respective digits to make the equation true.

TWO

x TWO

THREE

1 to 12 in a Cross

Write the numbers 1 through 12 in the squares below so that the two columns, the two rows, and the five squares that can be formed with four numbers in each have a total sum of 26.


Can you find the solution where no two consecutive numbers are next to each other horizontally, vertically, or diagonally?

1 to 19 in a Honeycomb

Can you place the numbers 1 through 19 in the honeycomb shown below so that there is a difference of at least 4 between any adjacent cells?

4 = 5

One of Professor Egghead's undergrad students had tried to prove that 2 is equal to 1, but he saw through that trick very quickly. A few days later the professor decided to turn the tables on his class. He brought in the following proof and wrote it on the chalkboard. He then told the class that the first student who could explain where the proof failed and why would get an automatic 'A' on the next test.


Given:

-20 = -20

1. Rewrite the two given values as differences:

16-36 = 25-45

2. Replace the four values from step 1 as products:

4² - (9x4) = 5² - (9x5)

3. Add 81/4 to each side:

4² - (9x4) + 81/4 = 5² - (9x5) + 81/4

4. Factor both sides of the equation:

(4 - 9/2)² = (5 - 9/2)²

5. Take the square root of each side:

4 - 9/2 = 5 - 9/2

6. Add 9/2 to each side and we end up with:

4 = 5

The students tried and tried to figure out where the problem was but none of them could solve the puzzle. Can you?

Seven Pairs of Numbers

If two 1s, two 2s and two 3s are arranged like this:

2 3 1 2 1 3

then the two 1s enclose 1 other digit, the two 2s enclose 2 other digits, and the two 3s enclose 3 other digits.


Can you find a similar arrangement using the seven pairs 1, 1, 2, 2,...7, 7?

The Joker

Four playing cards, one of each suit, and one each of Jack, Queen, King, and Ace are laid out in a row.

1. The Heart isn't next to the Club.
2. No card is next to its immediate senior in rank.
3. The colors of the suits alternate.
4. The King and Queen face in opposite directions.
5. The Jack of Diamonds is not in the row.

What are the four cards?

12 N on the F of a C

A reader of this page asks:

"I've been given a puzzle where I have to find words that fit in place of the single capital letters in this sentence:

There are 12 N on the F of a C.

The single letters are to be replaced with words that start with the provided letter."


I'm sure most of you can figure it out quite quickly. As for the reader who asked, if you try timing how long it takes to figure it out, I'm sure you'll figure it out a bit more quickly.

Longest Word with only One Vowel

A reader of this site wonders:

"Is there an 8 letter word with 1 vowel in the fourth position?"

In the English language, there is such a word, but it's not the longest word in the language with only one vowel. There is a nine letter word with only one vowel. To be fair, though, the eight letter word is a root of the nine letter word.

Feel free to venture your best guess in the comment section below.

Find the Hidden Message

There is a secret message hidden in the following array of numbers:

18526408428031762482422468044365750248

56348202842670462842246424262522528630

17928364505075849265296354889940124759

63409418743692236183670985232704962236

89264320091830485624510741061547354891

Can you discover the method that was used to create the numbers and decipher the code?

A E F H I K L M N

A visitor to this site asks:

"I have the first part to a series of letters but I don't know how to finish the sequence."

The sequence starts out as:

A E F H I K L M N

Can you find the last 6 entries in this series for our fellow reader?

Happy Birthday, Professor!

Some time ago, Professor Egghead invited 5 of his colleagues from the university to help him celebrate his birthday. Each of the 5 guests were given a special party hat and Professor Egghead wore one too.

After the candles were blown out and everyone had eaten cake, Professor Egghead stood up to make an announcement. "Each of us is wearing a unique hat selected from my sizable collection of hats I've collected over the past several years while travelling to Australia, Canada, Germany, Holland, and Japan. Curiously, the six hats we're wearing tonight are either red, yellow, or blue, and I can see that there is at least one of each color. As a special party favour I've set aside a special gift for the first person who can figure out the color of the hat on their own head without taking it off or looking into a mirror."

Professor Egghead's friends all knew he had a knack for picking out especially nice gifts for party favours so, of course, they each wanted to be the first to figure out the color of their hat. They each looked around the room at the hats on the heads of the professor and each of the other guests but they couldn't figure out the solution.

Suddenly all the guests stood up at the same time and exclaimed "I know the color of the hat on my head!"

They were all right and Professor Egghead, with a twinkle in his eye, gave each of them a gift that he had picked out especially for them.


Why was it so hard for Professor Egghead's guests to figure out what color hat was on their own head and how did they suddenly come up with the right answer?

Synograms

A synogram is a pair of words which are synonyms with the added rule that they are also anagrams (plus or minus one letter.)

For instance, 'bush' and 'shrub' are synonyms with only the letter 'R' added to 'bush' to make it an anagram of 'shrub'. Another synogram pair is 'ludicrous' and 'ridiculous'.

What other synograms can you find?

Oprowon Lumgalm

The writing you see below was discovered during an archeological dig that was sponsored by the university where Professor Egghead works. The scientists at the dig site were completely baffled by the strange writing--the letters looked similar enough to roman letters that they thought that the language must be related to the Indo-European group of languages but it was unknown to all who inspected it.

Professor Egghead asked if he might be allowed to inspect the artifact and completely deciphered the words in a matter of minutes. How quickly can you decode the text?

A Cabbage, a Tiger, and a Goat

Here's an old one, but a good one--it's probably the first logic puzzle anyone ever posed to me:


In this logic puzzle, a man is traveling with a tiger, a goat, and a cabbage. No one ever provided a convincing explanation for why he would be travelling with such a strange assortment--perhaps he was French. At some point in his journey the man came to a river which was too deep to wade across, and too wide to swim across so he was in a quandary on how to continue. He noticed a small boat tied to the near shore, but the boat was too small to hold all his belongings, but was large enough that he could safely row across with one belonging at a time. The problem he faced was that if he rowed across with the tiger then the goat would eat the cabbage, and if he rowed across with the cabbage, the tiger would eat the goat.

How could the man cross the river safely with all his belongings intact?

Barrels full of Pellets

You are presented with 10 barrels each full of small, round pellets. By looking at the pellets you cannot see or feel any difference between one barrel's pellets and the next barrel's pellets. You are told that 9 of the barrels contain 1 gram pellets and that the other barrel contains 2 gram pellets. The barrels are too large to move, so you can't try to push on them to see which seems heavier, but there is a weighing scale available to you. The problem is that for some strange reason you are only allowed to use the scale one time.

How can you figure out which barrel contains the 2 gram pellets using the scale only once?

2 = 1

Professor Egghead entered his classroom one morning when one of his undergraduate students boasted that he could prove that 2 is equal to 1. The student then showed the professor the following proof:


Given: x = 1 and y = 1 therefor:

x = y

1. Multiply each side by x:

x² = xy

2. Subtract y² from each side:

x²-y² = xy-y²

3. Factor each side:

(x+y)(x-y) = y(x-y)

4. Divide by the common term (x-y):

x+y = y

5. Put the initial values back in the equation:

1+1 = 1

or

2 = 1

Professor Egghead saw the problem right away, can you?

Had Had Had

Insert puctuation into the following sentence so that it makes sense:

"James while John had had had had had had had had had had had a better effect on the teacher."

Super Simple Sudoku

Below are 9 different Sudoku puzzles for your enjoyment. Unlike the Sudoku puzzles found in other places, these are each limited to the numbers 1 through 4, which makes them especially attractive for introducing Sudoku to young puzzlers or to the old crotchety folks who claim they "don't have time for such foolishness."

These simple puzzles are just right for elementary school teachers looking for games for their young students.

Knot or Not?

One day Professor Egghead was taking a walk in the park behind his house and spied a cord of some sort lying on the path. From the distance that he was from the rope he couldn't see if it was knotted or not.

As he continued to approach the cord he wondered about the probability of the cord being a knot.

In what orientations would the cord be a knot and what is the probability that the cord was knotted?

A Sequence of Numbers

Professor Egghead showed one of his graduate students the following 3 numbers:

1 5 9

and asked him "Do you know what the next one is?"

The student replied, "With only three numbers it would be very difficult to figure out the sequence but it appears to be an arithmetic series with each number being 4 more than the previous one."

Professor Egghead smiled knowingly and said "Some say the fourth number of the sequence is 3 times the second number, others claim it is the sum of the first three. Do you know the fifth, which is also the last number of the series?"

The student wrote four numbers on a piece of paper and was totally baffled as to what the last number could be. "Why would you think I should know this?" he asked.

Professor Egghead answered "Because you like to read."

Brothers and Sisters have I None...

Can you figure out who "this man" is in each of the statements below?

Brothers and sisters have I none, but this man is my son.

Brothers and sisters have I none, but this man's father is my son.

Brothers and sisters have I none, but this man is my father's son.

Brothers and sisters have I none, but this man's father is my father's son.

Brothers and sisters have I none, but this man's son is my son.

Brothers and sisters have I none, but this man's son is my father's son.

Brothers and sisters have I none, but this man's father's son is my son.

Brothers and sisters have I none, but this man's father's son is my father's son.

Shingles and Apples

There is an old story of a trader who put into Philadelphia with a boat load of shingles, some of which had been damaged in passage. He was asked by a Quaker merchant what the price was for the shingles.

"They are $10 a bundle," he replied, "if you choose the the bundles and $5 a bundle if I choose them."

The merchant thought for a minute and said, "Captain, I will buy your whole cargo, and you may choose the bundles."


The following puzzle involves a similar principle:

A man had an apple stand and sold his larger apples at three for a dollar and his smaller apples at five for a dollar. When he had just thirty apples of each size left to sell, he asked his son to watch the stand while he had lunch. When he came back from lunch the apples were all gone and the son gave his father $15. The father questioned his son.

"You should have received $10 for the thirty large apples and $6 for the thirty small apples, making $16 dollars in all."

The son looked surprised. "I am sure I gave you all the money I received and I counted the change most carefully. It was difficult to manage without you here, and as there were an equal number of each sized apple left, I sold them all at the average price of four for $1. Four goes into sixty fifteen times so I am sure $15 is correct."

Where did the missing $1 go?

The Dinner Party

Professor Egghead was talking to a colleague who told him about a dinner party he had attended with his wife. The colleague told him the following things about the party:

1. There were four married couples present.
2. Each person had a unique hobby.
3. The eight people were seated around a dinner table with the host and hostess at either end and three people seated on either side.
4. Only one married couple were seated beside each other.
5. A man was seated on either side of the hostess.
6. A woman was seated on either side of the host.
7. The hostess likes to ride horses.
8. Donna collects stamps.
9. Carol and her husband were seated on the same side of the table.
10. The piano player was seated next to his brother-in-law.
11. The person who grows roses was seated next to the person who does needlepoint.
12. Frank was seated directly across from the person who builds model planes.
13. Harold is seated to the immediate right of the hostess.
14. George and Betty were seated directly across from each other.
15. Alice is married to Edward.
16. Donna's sister-in-law is seated directly across from Carol.
17. The piano player was seated next to the hostess.
18. The stamp collector's husband was seated across from the model plane builder.
19. Carol was seated immediately to the left of George.
20. Alice is married to the fisherman.
21. The person who does needlepoint was seated across from the actress.
22. The actress was seated immediately to the right of the host.
23. The fisherman was seated across from his sister.
24. Frank was seated next to Edward.

What was each person's hobby, and where did they sit?