Batch 2 - Class 80 - Divisor Game, Chomp, Rook Game

Pre-Class Problem:

(Dudeney - 44) Mr. and Mrs. Jorkins have fifteen children, all born at intervals of one year and a half. Miss Ada Jorkins, the eldest, had an objection to state her age to the census man, but she admitted that she was just seven times older than little Johnnie, the youngest of all. What was Ada's age? Do not too hastily assume that you have solved this little poser. You may find that you have made a bad blunder!
Answer: 24 years (Little Johney is 3 years - its "seven times older" hence "eight times as old" - otherwise the answer would be 24.5 and 3.5)

Attendance: Anisha, Muskaan, Nandini, Liza, Abhiram, Arnav, Anishka

Class Notes:

Divisor Game

To play the divisor game, choose a number N and list all its positive divisors including 1 and N. Two players take turns crossing out a number from that list. Each time they cross out a number, they must cross out all its divisors. Whoever crosses out N is the loser of the game.

Let kids play some of the games.

Help organize - how should we draw the divisors and arrange them, so that crossing out is easier?
Lets take some simple cases. For example, product of two primes.
Do all these games have the same outcomes?Why? Does the outcome depend on the numbers or something else? Patterns?What about product of 3 primes?Take another pattern - an exponent of a prime number, and see what happens
Can we classify some patterns as winning patterns for the first person?
List down some of the patterns that are winning patterns for the first person
Can someone find a losing pattern for the first person?
(Teacher's Note: Make sure kids understand what is a "Losing Position" - its a position from where regardless of the move the person makes, she will lose - i.e. all moves take to winning positions). By contrast, "Winning Positions" are positions from where there exists a move to a losing position - i.e. there is at least one move to a losing position)Answer: 6 connected to 2 and 3 (no connecting 1, which has been taken out). Or a 12-4-2 and 12-6-3Note that all cases where there is a "corresponding move" on another leg qualify for this
How can any number be written in terms of patterns?
Think prime factorizations
Do you think the first player can always win?
Any ideas why? (Teacher's Note: If kids can get to the strategy stealing argument, great. Else given them a sense of it)There were some patterns where first person lost - why do those not show up?Lets try one "losing pattern" and see how we can avoid getting there in the simplest case.Thinking from the first person standpoint, can he leave the second person with one of these losing patterns? (again in simplest cases)

Chomp Game

Chomp is a simple 2-player game. There is a grid, and in the bottom-left corner of the grid is a poison square. Players alternate turns, and on each turn, the player selects a square and ’eats’ that square and all of the squares above and to the right of it (see example below). The player who is forced to eat the poison square loses.

Let kids play some of the games.

Do some versions of chomp game look like some positions of divisor game?
If you had square boards (3x3, 4x4, 5x5), who would win? How?
Can we use symmetry to come up with winning position?
Consider 2xn boards (2x3, 2x4, 2x5), who would win? How?
Identify some of the losing positions. Can we extrapolate from those to set up the second person into a losing position?Answer: Set up top row as one smaller than bottom row. Can you draw some parallels between this and the losing positions of Divisor game?
Note: 3xn case was solved in 2002 by a high school senior, Steven Byrnes, as part of a larger theorem.
Without necessarily thinking about how to solve the problem, can you come up with a proof that the first player can always win?
Think about the proof in Divisor game. Is there an analogous proof?
More explorations https://www.win.tue.nl/~aeb/games/chomp.html

Rook Game

In this game, a rook is placed in the bottom right corner of a rectangular board. Players take turns moving the rook either straight up any number of squares, or straight left any number of squares (but not both in one turn). The rook cannot move down or to the right. The player who places the rook in the top left corner of the board wins.

Play a few games against those around you on the boards below.

What are the winning positions for this game?
Hint: Use elements of symmetry to arrive at the same.

Homework

(Mott - 32) Mysterious Computation. Professor Digit claims that 161+134+145=503. Further, he also claims that 2625 when divided by 52 gives a quotient of 36 and zero remainder. Is he nuts?

References:

http://jrmf.org/problems/DivisorGame.pdf

https://mathcirclesofchicago.org/wp-content/uploads/2015/08/strauss.pdf

http://www.cemc.uwaterloo.ca/events/mathcircles/2015-16/Fall/Junior6_Nov4.pdf

http://www.math.cornell.edu/~mec/2003-2004/graphtheory/chomp/howtoplaychomp.html

https://ia902701.us.archive.org/4/items/AmusementsInMathematicspdf/AmusementsInMathematics.pdf - Dudeney

Mathematical Puzzles, Geoffrey Mott-Smith