A Strange Safari - Part 2

Earlier in the month we posted a puzzle about Professor Egghead's Australian safari where he, sadly, did not see any animals other than horses and people on his first day.  Luckily the professor did get a chance to see animals on his second day of safari.

After the second day Professor Egghead called me again and this time he was happy to announce that he had seen real wildlife while out riding the horses.  I asked him how many animals he had seen and, again, he didn't remember the exact number of all the animals, but he did remember that, including the horses and people from the day before, there were a total of 52 eyes and 74 legs.

If there were twice as many 4 legged dingoes as there were 2 legged ostriches, how many of each wild animal did the professor see?


(You'll have to be a bit creative to get the right answer to this one.)

The Professor's Wooden Measuring Stick

Professor Egghead was accompanying a colleague on a remote archaeological expedition.  On the first day of the dig, however, he was surprised to discover that he had forgotten some of his tools at home.  Most important of his missing tools was his ruler, used for measuring lengths of small objects uncovered at the dig site.  He was able to find a straight, blank wooden stick that measured exactly 10 inches and borrowed a ruler from one of the other members of the team to make marks on his stick to create a ruler of his own.  To keep things simple the professor wanted to make as few marks on his new ruler as possible.

What is the fewest number of marks the professor could make on the stick so that he could still accurately measure any whole number length from 1 inch to 10 inches?  For instance, placing a mark 1 inch from one end would allow the professor to measure 1 inch and 9 inches.

A Strange Safari - Part 1

Professor Egghead went on a safari to Australia last January to see ostriches and dingoes. He called me up the day he arrived on the ranch to tell me what it was like. I asked him how many wild animals he had seen and he replied that he had only seen people and horses, but he didn't remember how many of each there were. He did remember, though, that between the people and horses there were 22 eyes and 34 legs.

How many humans and how many horses did the professor see?

Professor Egghead's Cuckoo Clock

On one of his trips to Switzerland Professor Egghead bought a handcrafted cuckoo clock that chimes on every hour and half hour mark. On whole hour marks the little birdie cuckoos once for each hour, and on half hour marks it cuckoos only once.

One night our favorite professor was wakened suddenly and realized that the clock had chimed but he did not know how many times. As he lay awake thinking about nothing in particular he heard the birdie cuckoo once and he started to wonder what time it was.

What is the longest amount of time Professor Egghead would have to lie awake before he knew for sure what time it was?

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?

I Like Ike Calendar

In 1952, Professor Egghead's father voted for Dwight Eisenhower for US President and had kept the calendar as a keepsake. Recently, Professor Egghead was going through some of his father's keepsakes and discoverd this calendar from 1952 and noticed that all the dates in 1952, which was a leap year, coincided exactly with the dates in 2008, which is also a leap year.

This made the professor wonder, how many different calendars would you need to have in order to represent every possible combination of yearly calendar?

Those Incredible Colored Socks

Professor Egghead woke up one morning and started dressing for a day in the office. He reached into his sock drawer and felt that there were four individual socks. He knew that two of those socks were black, one was white, and one was green. He grabbed two at random and put them on before leaving for work.

If one of the socks he was wearing was black, what is the probability that the other one was also black?

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.

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?

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?

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?

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?

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?

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?

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."