To get the kids started, if they have the time, ask them to think about the following - In a 24 hour period, how many times does the hour and minute hands of the clock overlap? If they get that, ask them to find out how many times the hour, minute and second hands overlap all together. If they are unable to do it, no problem. If they want to not just think about it, but do it with a real clock, please allow!
(MartinShCol - 4.17) How many times in 24 hours do the hour and minute hands overlap - Answer 22. Starting from 12 noon, the next overlap is at 1 hr and 5+ minutes, the second at 2 hr and 10+ minutes and so on. Thus, the overlap is not every 60 minutes, but about every 65 minutes. Hence in a 12 hour period, there are only 11 overlaps and not 12. Mathematically, in the duration that the hour hand covers x distance, the minute hand covers 12 x distance, and hence the difference is 11x. This difference is an integer number of revolutions since relative to hour hand the minute hand has moved integral number of revolutions more, i.e. 11x should be integral. There are 11 solutions to this (1/11, 2/11 and so on. Note that 0/11 and 11/11 correspond to 12 noon and hence are the same solution), each corresponding to one overlap.
How many times would the hour, minute and second hands all overlap - Answer 2. Like above, hours and seconds overlap at integer values of 719x. Since 11 and 719 are both prime, there are only two values of x, i.e, 0 and 1 that satisfy both equations (both correspond to 12 noon and midnight respectively)Problem for parents: How many times do minutes and seconds hand overlap in 24 hours?
Instructor Note: Lead through actual clock and kids working through it
(Moscow - 88) There are 9 kids in a circle. Count clockwise, incrementally 1, 2, 3, 4...10. At each counting, eliminate the person at that position and restart counting from next person onwards. The last survivor wins. Where would you start the counting so that you win? What if there are 10 kids?
Homework: Take A, 2, 3,...10, J, Q, K from a deck of cards. Arrange them in a pile, such that 1st top card is "1", then put the next top card at bottom, and second card should be "2", then put next 2 cards one by one at bottom and third card should be "3", and so on.
Instructor Note: Lead through an example, and then draw the translation to "x persons ahead", or "x persons after"
(Contributed by Tara) Truthtellers and liars. There is a village of truthtellers (who always tell the truth) and liars (who always lie). I want to go to the village and reach a fork in the road, where a villager is standing. Can I ask one question to the villager, without knowing whether she is a truthteller or a liar, which will allow me to determine which path to take on the fork to reach the village
Answer 1: "If I asked you which road goes to the village, what would you say?" (Take the road the villager points to. Logic is Truth->Truth = Truth, and Lie->Lie = Truth)Answer 2: "If I asked a member of your opposite tribe the road to the village, what would they say?" (Take the road the villager does not point to. Logic is Truth->Lie = Lie, and Lie->Truth = Lie)
Instructor Note: Lead through Lying about a lie may give a truth; demonstrate through a non-ace card, asking "is this an ace" and "if I asked you if this is an ace, what would you say"
(MC - 0 - 2) Ann, John, and Alex went to Disneyland with ticket costing $5 per head (yes, Indian kids wanted dollar pricing!). But they have only $10, $15, and $20 chips. How do they pay for the tickets.
Answer: Ann pays $15 and buys three tickets. John and Alex give Ann $20 chips, and get $15 back each.Rejoinder: What if Ann has only $10 chips, John has only $15 chips, and Alex has only $20 chips.
Instructor Note: Lead through total amount to be paid ($15), getting that done. Now make sure each person has the net total they should have
(Contributed by Manas) There are four people who need to cross a bridge. They take 10, 5, 2, and 1 minutes respectively to cross a bridge. There is only one light and anyone crossing the bridge must use it. Maximum of two people can cross the bridge at any time. What is the minimum time in which all four people can cross the bridge?
Answer: 17. 2 and 1 cross, 1 comes back, 5 and 10 cross, 2 comes back, 1 and 2 cross
Instructor Note: If students get part of puzzle right, accept that, and then focus on the subpart which is not solved. Ask students to list down all possible combinations.
(Shakuntala - 23) How many squares does a 3x3 grid have (Answer: 14). How many do 1x1, 2x2, 3x3, 4x4, 5x5... grids have (Answer: 1, 5, 14, 30, 55, ...). What is the pattern in these numbers? (Answer: Sum of squares of first n numbers). Why do we land up with such a pattern (Answer, counting from largest to smallest square, there are 1, 2, 3, 4, 5... choices on how to select an edge of that size of square square on each side of the grid)
Instructor Note: Kids tend to be able to answer the sequence. For pattern, walk through number of largest squares to smallest in a given grid, and then prompt students to identify the pattern.
(MC - 0 - 4) There are 24 kg of wheat in a sack. There is a weight balance (but no weights) available. How do you weigh out 9 kg (Answer: Split 24 into 12 and 12; then 12 into 6 and 6; then 6 into 3 and 3 - give the remaining 6 and one 3 pile)
Instructor Note: Kids tend to get this. If not, ask them to do the only thing they can do at each step
(Shakuntala - 197) A frog is in a 30 ft well. At every hour, it climbs up 3 ft, but slips back 2 ft. How many hours does it take for the frog to climb out (Answer: 28)
Instructor Note: Lead in the kids with 5 ft well and if required, a 3 ft well
Joke: Circle 6 numbers that total to 21 in the following grid (Answer: First three and last three inverted)
9 9 9
5 5 5
3 3 3
1 1 1
Instructor Note: After a while tell that its a joke. Use a whiteboard you can turn upside down.
References: More Puzzles, by Shakuntala Devi
Mathematical Circles (Russian Experience), by Dmitri Fomin, Sergey Genkin, Ilia Itenberg
The Moscow Puzzles, by Boris A. Kordemsky
The Colossal Book of Short Puzzles and Problems, by Martin Gardner