Arithmetic Puzzles 01

Here are a small collection of technical puzzles that I find interesting. The answers to some of them may elude me from time to time. Their appeal to me lies in the fact that one can spend MANY hours on them and get nowhere. Sometimes the solution is elegent and sometimes ugly, so as Erdos would say, if you have a "book" proof for some of these problems, please do tell.

1. A Trig Inequality: Cos(Sin x) > Sin(Cos x)

2. The Monotonic Subsequence: Consider any permutaiton of {1,2,...n^2+1}. There must exist a monotonic sub-sequence of size at least n+1.

3. A Probability Problem: Let X and Y be two independent identically distributed real random variables. Show that
P[X-Y<=2]<= 3 P[X-Y<=1]

4. Divisors:

a} How many numbers less than 1000 are not divisible by 3 or 5 or 7? Generalize.

b}How many divisors does 9112001 have (including 1 and 9112001) and what is the sum of all these divisors?

5. Co-Prime Probability: Given two positive numbers, what is the probability that they are relatively prime? (Assume a uniform distribution) Generalize to K numbers.

6. A Line Through 2 Points: Given any set of greater than 2 points on the plane, not all on a line, there is at least one line that passes through exactly two points.

7. Matching the Hats: 100 coats are randomly distributed to 100 owners. What is the probability that at least one owner gets his own coat?

8. Betting the Deck: A deck has 26 red and black cards. A red card rewards you $1, and a black card fines you $1. You draw cards one by one, and may stop whenever you wish. What is your optimal strategy?

9. If x is a positive rational number, show that x^x is irrational unless x is an integer.

10. Let H(n) = 1/1 + 1/2 + ... + 1/n. Show that, for n > 1, H(n) is not an integer.

Logical Puzzles 001

Puzzle 1. HARE & TORTOISE

Haretown and Tortoiseville are 63 miles apart. A hare travels at 7 miles per hour from Haretown to Tortoiseville, while a tortoise travels at 2 miles per hour from Tortoiseville to Haretown.

If both set out at the same time, how many miles will the hare have to travel before meeting the tortoise en route?



Puzzle 2. THE CARPENTER AND THE NAIL

A 23" x 23" square metal plate needs to be fixed by a carpenter on to a wooden board. The carpenter uses nails all along the edges of the square such that there are 24 nails on each side of the square. Each nail is at the same distance from the neighboring nails. How many nails does the carpenter use?



Puzzle 3. THE BIRD AND THE TRAIN

The distance between Station Atena and Station Barcena is 78 miles. A train starts from Atena towards Barcena. A bird starts at the same time from Barcena straight towards the moving train. On reaching the train, it instantaneously turns back and returns to Barcena. The bird makes these journeys from Barcena to the train and back to Barcena continuously till the train reaches Barcena. The bird finally returns to Barcena and rests. Calculate the total distance in miles the bird travels in the following two cases:

(a) the bird flies at 70 miles per hour and the speed of the train is 60 miles per hour
(b) the bird flies at 60 miles per hour and the speed of the train is 70 miles per hour




Puzzle 4. THE GREAT PIZZA DEAL

The 8" pizza sells for $ 5.99 at my favorite pizza store. The store claims they have a great deal on the large 14" pizza, which is specially priced at $ 16.5. What is the per cent discount the store is offering?



Puzzle 5. THE GAME OF CHOCOLATE

Last vacation, my cousin came over to stay at my home. We made the most of her stay at my place... and I even earned a few chocolates.

Everyday, we would play a game of chess. Whoever lost the game owed a chocolate to the other. After the last game we played (that was the day she was to leave), we counted the number of games each of us had won and lost. Wow! I had won more than her. So, she handed me 16 chocolates... though she herself was the winner in 9 games.

How many days did my cousin spend at my place?




Puzzle 6. GRANDPA'S AGE

"My grandson is about as many days as my son is weeks, and my grandson is as many months as I am in years. My grandson, my son and I together are 120 years. Can you tell me my age in years?"



Puzzle 7. CRITICAL CHAIN

The son of a rich bullion merchant left home on the death of his father. All he had with him was a gold chain that consisted of 123 links. He rented a place in the city center with a shop at the lower level and an apartment at the upper level. He was required to pay every week one link of the gold chain as rent for the place.

The landlady told him that she wanted one link of the gold chain at the end of one week, two gold links at the end of two weeks, three gold links at the end of three weeks and so on.

The son realized that he had to cut the links of the gold chain to pay the weekly rent. If the son wished to rent the place for 123 weeks, what would be the minimum number of links he would need to cut?



Puzzle 8. PYRAMIDS OF EGYPT
My Dad has a miniature Pyramid of Egypt. It is 6 inches in height. Dad was invited to display it at an exhibition. Dad felt it was too small and decided to build a scaled-up model of the Pyramid out of material whose density is (1/ 10) times the density of the material used for the miniature. He did a "back-of-the-envelope" calculation to check whether the model would be big enough.

If the mass (or weight) of the miniature and the scaled-up model are to be the same, how many inches in height will be the scaled-up Pyramid? Give your answer to two places of decimal.



Puzzle 9. GIVE AND TAKE

Glenn and Jason each have a collection of cricket balls. Glenn said that if Jason would give him 12 of his balls they would have an equal number; but, if Glenn would give Jason 12 of his balls, Jason would have 4 times as many balls as Glenn. How many balls does Jason have?


Puzzle 10. UNIQUE NUMBER
There is a number that is 9 times the sum of its digits. What is this number?


ANSWERS: (1)49 miles (2)92 (3)(a) 91 (b) 72 (4)10% (5)34 (6)72 (7)4
(8)12.93inches (9)52 (10)81