Batch 3 - Class 201 - Game of Cups
Pre-Class Exercise
- A 10x10 square is decomposed into exactly 75 squares of various integer sizes. How many 3x3 squares are there in the decomposition?
- Answer: 2. One way to think about it is that any 2x2 square eats up 3 unit squares (it has 4 squares, and 1 would anyway be there in a 1x1 square) over a unit square, 3x3 eats up 8, 4x4 eats up 15, 5x5 eats up 24 and so on. The total "eat count" must be 25. This means that there are no 4x4 or bigger squares, because then no combination of larger squares eats up exactly 25 squares. It also means there must be 2 3x3 squares, because with other values, we can't add up enough 2x2 squares to total exactly 25 "eaten" squares.
Attendance Aashvi, Siddhant, Anishka, Arnav, Rehaan, Ayush, Kushagra, Kabir, SiddharthM
Class puzzles
Cups is a game invented in 1965 by father-son duo of Arthur and Wald Amberstone. It is a contemporary two-ranked single-lap member of the ancient game family of Mancala.
The board is constructed of ten containers: eight small ones called cups and two large pots. 80 beans are needed. Each player has four cups in front and a pot at end of row on far right. Each player receives 40 beans as stock.
Players only play on their own four cups and pot. In their turn, each player can
- Play 1,2,3 or 4 beans from their stock and sow them from her leftmost cup towards the pot. Sowing means to place them in one by one along the line of cups. In this case, if the last bean lands in an empty cup, the opponent's cup adjacent to the said empty cup is emptied (Capture) and those beans are put in the capturing player's pot.
- Empty one of the cups and sow the contents towards the pot. This can be done only if number of beans in the cup is exactly enough to reach the pot (last bean has to land in the pot)
If a cup has more beans than required to perform option 2, then sowing from that cup becomes impossible. Avoid this situation.
Object of the game is to have more beans than opponent in your pot at end of game. The game continues till either player can make a move.
Variations: You can play with fewer or more cups. Beans provided are ten times the number of cups with each player.
Homework
- (MartinShColl 9.20) There is a cake with square top and four sides frosted. We want to cut the cake into 7 pieces, so that each piece has the same volume and same area of frosting.
- Answer: Divide the square perimeter into seven equal segments and cut vertically. The perimeter acts as base of triangles to calculate the top surface area of frosting, and hence equal (the side frosting is equal because base and height are same). The volume is the top surface area multiplied by height, and hence again equal.
References:
The Colossal Book of Short Puzzles and Problems, by Martin Gardner