# Nick's Mathematical Puzzles: 21 to 30

## 21. Birthday line

At a movie theater, the manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket.  You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout a 365 day year, what position in line gives you the best chance of being the first duplicate birthday?

`Hint  -  Answer  -  Solution`

## 22. Isosceles angle

Let ABC be an isosceles triangle (AB = AC) with BAC = 20°.  Point D is on side AC such that DBC = 60°.  Point E is on side AB such that ECB = 50°.  Find, with proof, the measure of EDB.

`Answer  -  Solution`

## 23. Length of hypotenuse

Triangle ABC is right-angled at B.  D is a point on AB such that BCD = DCA.  E is a point on BC such that BAE = EAC.  If AE = 9 inches and CD = 8 inches, find AC.

`Answer  -  Solution`

## 24. Die: median throws

What is the minimum number of times a fair die must be thrown for there to be at least an even chance that all scores appear at least once?  (Computer assistance advisable.)

`Hint  -  Answer  -  Solution`

## 25. Die: mean throws

What is the expected number of times a fair die must be thrown until all scores appear at least once?

`Hint  -  Answer  -  Solution`

## 26. Two packs of cards

Players A and B each have a well shuffled standard pack of cards, with no jokers.  The players deal their cards one at a time, from the top of the deck, checking for an exact match.  Player A wins if, once the packs are fully dealt, no matches are found.  Player B wins if at least one match occurs.  What is the probability that player A wins?

`Answer  -  Solution`

## 27. 1000th digit

What is the 1000th digit to the right of the decimal point in the decimal representation of (1 + )3000 ?

`Hint  -  Answer  -  Solution`

## 28. Making 24

Using only the numbers 1, 3, 4, and 6, together with the operations +, −, ×, and ÷, and unlimited use of brackets, make the number 24.  Each number must be used precisely once.  Each operation may be used zero or more times.  Decimal points are not allowed, nor is implicit use of base 10 by concatenating digits, as in 3 × (14 − 6).

As an example, one way to make 25 is:  4 × (6 + 1) − 3.

`Hint 1  -  Hint 2  -  Answer`

## 29. xx

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

`Hint  -  Solution`

## 30. Two pool balls

A cloth bag contains a pool ball, which is known to be a solid ball.  A second pool ball is chosen at random in such a way that it is equally likely to be a solid or a stripe ball.  The ball is added to the bag, the bag is shaken, and a ball is drawn at random.  This ball proves to be a solid.  What is the probability that the ball remaining in the bag is also a solid?

`Answer  -  Solution`