There is a tournament with 32 players. In each game, two people play and the winner moves to the next round. In the next round, again two people play and the winner goes to the next round, and so on, till there is one overall winner. How many matches will be played in the tournament?
There is a tournament with 32 players. In each game, two people play and the winner moves to the next round. In the next round, again two people play and the winner goes to the next round, and so on, till there is one overall winner. How many matches will be played in the tournament?
Answer: 31
Instructor Note: Illustrate with beads. If kids are not able to work with 32, ask them to solve for 2, 4, 8 and see the pattern. Once they get powers of 2, ask them to solve for 5 participants, and ask why the n-1 pattern always holds. Guide them to the solution focusing on eliminations rather than counting all matches
Lateral thinking warmup: (FirstMensa - 13) - Position three coins in such a way that two heads are completely to right of a straight line and two tails are completely to the left
Instructor Note: Give kids coins to do this with
A king had a daughter who wouldn't smile. The king announces that whoever will make the daughter smile will get whatever they want. A wise man came and made the princess smile. And he asked for? He asked for rice over next 365 days - 1 grain the first day, 2 grains the second day, 4 grains the third day, and so on. The king agreed. The man came back on the 365th day to collect his rice. And as they started counting, the king could not fulfill his promise. Why? (Let kids assume 1000 grains of rice = 1kg)
Instructor Note: Idea is to let kids appreciate how fast binary sequences (introduce the term later) explode. 1 kg on 10th day, 1000 kg on 20th day, 1M kg on 30th day, and so on. Mass of earth = 6 x 10e24 kg. Mass of universe 10e53 kg.
As followup, ask kids to doodle binary trees (Khan Academy Videos) to get a first hand feel or how fast this grows. Ask for volunteer first on basis of time bids, and then let them realize the magnitude of underestimation. Crashing binary trees -> Fractals.
(MC Chap 0, Prob 1) - There is a jar of yeast and the yeast doubles every day. It fills the jar on 30th day. When was the jar half full?
Answer - Day 29
Instructor Note: Idea is for kids to appreciate that binary numbers grow really fast, and (optional) they add as much in the last step as all of previous steps
Lateral thinking warmup: (FirstMensa - 43) - Four pennies at four corners of square. Change position of one to produce two straight rows with three counters each
Instructor Note: Give kids coins to do this with
(Manas Contributed) - There is a rabbit which is some distance away from pizza slice. In each step, the rabbit covers half of the remaining distance to the pizza. In how many steps does the rabbit reach the pizza?
Answer: Never
Instructor Note: Illustrate that a finite distance can have infinite steps
(Shakuntala - 92) A ball is dropped from height of 8 ft and bounces off back to half the height every time. What is total distance before it comes to rest
Answer: 24
Instructor Note: Illustrate with a ball. Idea is to familiarize with sequences and the notion that infinite sequences can have finite sums. Followup: Familiarize students with a series 1+1/2+1/4+1/8. Show the sum of circles in triangle visualization, from Khan Academy. Use the circles visualization to motivate another sequence, such as 1+1/3+1/6+1/9. In comparison to last problem, illustrate that infinite steps can lead to finite distance, and in multiple ways (use circles visualization)
(MC Chap 0, Prob 19) - Distribute 127 one dollar bills among 7 wallets so that any integer sum from 1 to 127 can be paid without opening a wallet.
Answer: Binary numbers 1, 2, 4, 8, 16, 32, 64
Instructor Note: Illustrate with beads with smaller number say 31. Idea is to motivate an alternate counting notion (dont press too hard on explaining binary math but introduce if kids feel comfortable)
Joke - fastest fingers first - (MartinShCol - 4.1) - if you picked 3 apples from a basket containing 13 apples, how many would you have?
Answer: 3
Fibonacci Rabbits: Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?
Answer: 144
Instructor Note: Idea is to introduce Fibonacci number. Also walk them through fibonacci spiral.
Count the spirals on a pine cone
Homework: Bring one example of Fibonacci sequence from nature
(Mount Fuji) - Ask your parents! There is a rectangular cake, out of which a rectangular piece has been arbitrarily cut out. Of the remaining cake, cut it into two equal halves with a single straight cut
Answer: Cut through the line passing through the center of original cake, and center of the cutout
Instructor Note: Walk kids through by asking them how many ways could they cut the original cake into two equal halves, if the cut out was not there. What is common in those cuts? Then introduce the cutout.
References: More Puzzles, by Shakuntala Devi
Mathematical Circles (Russian Experience), by Dmitri Fomin, Sergey Genkin, Ilia Itenberg
The Colossal Book of Short Puzzles and Problems, by Martin Gardner
A First Mensa Puzzle Book, by Philip J Carter, Ken Russell
How do you move Mount Fuji?, by William Poundstone