A Fruity Problem

Professor Egghead's nephew spent a summer working as a stockboy for a large grocery store. It was a rather unusual grocery store, however, and they had a very strange way of arranging the fruit in the fresh produce area.

In one group you would find any of the following fruits:
apple, banana, grape, and orange.

In a different area you would find fruits like:
mango, nectarine, peach, and pear.

One day a new crate of fruit arrived and the young man wasn't sure where to put it. Where would you put a crate labeled strawberry?

A Different Kind of Sudoku

Here's a sudoku challenge of a slightly different sort. In the 5x5 grid shown below, write (x,y) pairs, with x and y ranging from 1 to 5 inclusively, with the stipulation that x and y values cannot be repeated in any row, column, or diagonal.

For instance, if you write the (x,y) pair (1,3) in the top left cell, then you can not have any other pairs with an x value of 1 or a y value of 3 in the top row, the left-most column, or the main diagonal that runs from the top left to the bottom right.

This puzzle can be solved fairly quickly and is not quite as difficult as it may seem.


Click on the picture for a larger version

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.

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?

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?

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?

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.

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?

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.

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."

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?