## Always remember: e~3.14159i=-1.

Can you comprehend the concept of infinitude, mortal? Do you see mathematical order in what seems to be chaos? Can you juggle numbers, seven at a time, with flaming torches, whilst cooking an egg, on stage, live, in fabulous Las Vegas? Er, wait a minute; never mind that last ominous query. Anyway, here's the number puzzles... enjoy. Pencil and paper might be helpful, but for most of these, they shouldn't be necessary.

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## The Number Puzzles

On a barnyard, there was a chicken, a turkey, and a crow. The chicken could fly at 5 mph, but, as chickens are poor flyers, he soon tired. After only 2 minutes of flying, he had to walk at 1 mph for three minutes before he could fly again. The turkey could fly continuously at 4 mph, but he had a habit of swerving back and forth, and so for every 3 feet he flew, he actually only travelled 2 feet. The crow could fly straight, at 2 mph.

Which animal would win a one-mile race?

Joe walks into a stamp store with a stamp worth \$30. Inside, the dealer gives Joe a stamp worth \$16 and a stamp worth \$14 in trade. Joe then leaves with his two stamps. What has happened to the average value of a stamp in the stamp store: has it gone up, gone down, stayed the same, or is it impossible to tell?

What number is missing in this sequence?

`3  ?  4  1  5  9  2  6  5  3  5`

What number is next in this sequence?

`34  21  13  8  5`

You have 120 oranges, and three orange bins. Each bin can hold 40 oranges. You must put all of the oranges in the bins. Once this is done, you will write the number of oranges in each bin on its lid.

How many different arrangements of numbers on the lids are possible?

Did you find the last one too easy? This is the same problem as above, but each bin can hold up to 60 oranges. You may elect not to put any oranges in a bin.

Now how many different arrangements of numbers on the lids are possible?

On the road to Canterbury, four pilgrims met. Their names, in no particular order, were John, Geoffrey, Martin, and Stephen. This is the conversation they had on the road:

John: "I usually only tithe two pence, but if I beat Stephen to Canterbury, I'll gladly tithe double!"
Martin: "If I get to Canterbury first, I shall show my gratitude by tithing six pence! If I don't, I shall tithe 4 pence anyway."
Stephen: "You wayward fools are tossed by the wind! No matter what happens before here and Canterbury, I shall tithe three pence only. Heaven forgive your false piety!"
Geoffrey: "Shut up, Stephen. I pledge three pence if I beat you to Canterbury, but nothing if I don't. We'll see who's right!"
Everybody was true to his word, and a total of thirteen pence was tithed at Canterbury. In what order did the pilgrims arrive?

Let n be a positive integer greater than 3. Which of these series, when applied to n, will yield the square of n, with the last digit truncated?

(n - 3) + (n - 6) + (n - 13) + (n - 16) + (n - 23) + (n - 26)...
(n - 3) + (n - 7) + (n - 13) + (n - 17) + (n - 23) + (n - 27)...
(n - 3) + (n - 6) + (n - 9) + (n - 12) + (n - 15) + (n - 18)...
(n - 3) - (n + 7) + (n - 13) - (n + 17) - (n - 23) - (n + 27)...
Each series is carried out until no more positive terms can be added.

1 January 1989 was a Sunday. What day of the week was 1 January 1988?

Four leprechauns, under the orders of Queen Lucretia, met in a fairy-ring deep in the forest one day. This fairy-ring was perfectly circular, and had a radius of 1 glee-hop. One leprechaun, starting at the center, moved due north. When he came to the very edge of the fairy-ring, he magically planted a new, smaller fairy-ring, with a radius of 1/2 of a glee-hop. This new fairy-ring was perfectly adjacent to the old fairy-ring. He then continued north and made a fairy-ring of radius 1/4 glee-hop that touched the 1/2 glee-hop ring, and then a 1/8 glee-hop fairy-ring that touched the 1/4 glee-hop fairy-ring, and so on, forever. (These leprechauns really enjoyed their work.) As this leprechaun did this, the other three leprechauns did the same thing, with one going south, one going east, and one going west. (If the sun got in the eyes of the east or west leprechauns, they would occasionally switch places. As this was deep in the forest, though, this rarely happened.)

Anyway, as the leprechauns finished (?) they had created a figure, made up of fairy-rings of various sizes, and perfectly symmetrical about the north-south line, east-west line, and about a point in the center. Does this infinite number of fairy-rings, collectively, have a finite area? If so, what is the area (in square glee-hops, please!) How far has each leprechaun walked? (That is, what is the radius of this figure?) No calculus is required.

You can construct a number from any date of the year by adding the number of the month with the number of the day. For example, April 17th would become the number 21, since April is the fourth month, and 4 + 17 = 21. November 23 would become 34, and so on.

How many different numbers can you make, using dates of the Gregorian calendar?
Since date representation by this method is not unique, what are the most frequent numbers?
What are the most infrequent numbers?

Here's something for those mathematically-inclined:

Let F(x) be a function defined for positive integers greater than or equal to 2. It is equal to "x" multiplied by the number of terms in the prime factorization of "x" - F(4) = 8, since 4 = 2 X 2 (2 terms), F(30) = 90, since 30 = 2 X 3 X 5 (3 terms).

What is the maximum value of F(x) on the interval [2, 50000], and at what value of "x" does it attain its maximum?

Let's play a little game, somewhat reminiscient of the old pencil-and-paper game, "Cookie". You start with a number square, so:

```1 0 4
3 2 1
1 3 5
```
...and you wish to make every square equal to 1. You may pick any number in the square and deduct 2 from it, but every number above it or to the left of it is incremented. Hence, if you take 2 away from the upper-left square, no other squares are altered, however, if you take 2 away from the lower-right square, every other square has 1 added to it.

What's the minimum number of moves to win this game?