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.
Get three people to stand in a row with cards numbered 1-9 with each (with no card being 0), and ask them to count. Ask them to come up with the rules of counting:
Person at the left most position always increments his card (after 9, comes 0)
Any person after that increments their card when the card to the left goes to 0
Notion of units, tens, hundreds place. The number formed at any point is units place, plus 10 times tens place, plus 100 times hundreds place
3 people can count to 10^3, 4 will be able to count to 10^4 and so on
Binary Counting System
Lets count binary - Get 5 people to stand in a row with cards numbered 1 (and reverse side being 0), and ask them to count. Rules of counting same as decimal system, but now 10 replaces by 2
Person at the left most position always increments his card (after 1, comes 0)
Any person after that increments their card when the card to the left goes to 0
Notion of units, twos, fours, eights, sixteens place. The number formed at any point is units place, plus 2 times twos place, plus 4 times fours place and so on
5 people can count to 2^5, and so on
Now give them cards numbered 1, 2, 4, 8, 16 and do the same counting, to show how those numbers add up. Play a counting game where they count to 31 using those numbers. See http://csunplugged.org/binary-numbers
Motivate the notion of binary as ON or OFF. Provide linkage to how computers work - how are 0 and 1 stored (as charged particles)
Exercise: Give each kid a number that they must write using binary notation
MartinShCol - 1.10 (Pohl's proposition - binary math combinations) - If I write on the board, a number of zeros representing the number of binary counters and hide the row of zeros, you need to write how many numbers I can represent using those counters
Answer: Put a 1 before my row of zeros. The resulting number is the answer in binary notation
Game Play
Homework: On a 8x8 chess board, there is a lame tower, that players can successively move either any number of steps to the left, or any number of steps down. Figure out a winning strategy. When can you win as the first player.
Instructor Notes: Make groups of two and let them play the game. If someone gets it early, dont have them tell the solution. Let others discover. After some time, have people start marking spots on a common board, where they think the first player always wins, see if concentration develops.
Homework: Ternary Counting System
Write 1-30 in Ternary (base 3) counting system. Hint, rules stay as before:
Person at the left most position always increments his card (after 2, comes 0). Each place can have values 0,1,2
Any person after that increments their card when the card to the left goes to 0
Notion of units, threes, nines place. The number formed at any point is units place, plus 3 times threes place, plus 9 times nines place and so on
Homework Problem: 9 balls, one is lighter. Find which in two weighings
References: The Colossal Book of Short Puzzles and Problems, by Martin Gardner