Batch 3 - Class 222 - Game of Patterns
Pre-Class Exercise
- (Geoffrey - 161 Modified) The sketch below shows the plan of industrial establishment with secret weapons. There are only three gates which are guarded. There is some information leakage, and they found this note supposedly written for the spy
"Meet me every tuesday at the corner near my office. Enter by northwest gate and take a different route each time. That gives you 210 choices".
The detectives presumed the spy would enter top left and would always move to the right or down, in order to be efficient. Which office occupant was leaking the information?
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- Answer: this is same as ways of arranging a number of 1's in a bit string. Since 210 appears as 5th number in 11th row, it must be at intersection of 7th and 5th Street (there is only one such combination above - if it was a larger grid, there would be two possibilities)
Attendance: Kabir, Aarkin, Vansh, Advay, Kushagra, Shikher, Ayush, Angad, Rehaan
Class puzzles (Repeat from Class 146)
Game of Patterns
(Mathematical Circus - Chapter 4)
Game of Patterns is an inductive game (Induction means going from specific to generalization) on a 6x6 grid. There is a designer, designs the nature, who puts a pattern with four symbols on the 6x6 grid. Example patterns below:
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Other players have empty 6x6 grids. They can mark any number of squares with a small diagonal line in corner, and "ask" the designer about the symbol in those cells. They can do so any number of times, in no particular order. At any stage then, they can guess the symbols in all other cells, and give it to the designer for verification. This is like trying to figure out the order of nature. Finally, the scoring is as follows:
- Each person gets a score +1 for each correct guess, and -1 for each incorrect guess. The guesser may also drop out at any point, with zero score.
- The designer gets points basis the best and worst score of all guessers. The designer's points are (points of best guesser - points of worst guesser). Further, for each guesser who has dropped out, the designer loses 5 points for the first one, 10 for the second one and so on.
The above scheme ensures that the pattern should be such that some guesser can do very well at it, and someone else may do very poorly at it, but without dropping out.
Notion of strong conjecture - something that is easy to falsify, such as "all cells have stars". Weakest conjecture, hard to falsify, such as "any cell can have any symbol".
Let kids play the game and explore what kind of patterns work well. Rotate the role of Designer by throw of dice.
Homework:
- Five school girls were weighed in pairs and found to be 129, 125, 124, 123, 122, 121, 120, 118, 116 and 114 pounds. How much do they weight individually?
- Answer: 56, 58, 60, 64, 65. Reasoning below
- All girls must weight different, otherwise there would be repetition in pair weights. Let weights be a>b>c>d>e. Then largest weight must be a+b, then a+c. Lightest must be e+d, and then e+c. Sum of all the pair wise weights is 4(a+b+c+d+e) = 1212, and hence (a+b+c+d+e) = 303. Now c = (a+b+c+d+e)- (a+b) - (d+e) = 1212 - 129 - 114 = 60. a+c = 125, and hence a = 65, and a+b = 129, so b = 64. Similarly the rest.
References:
Mathematical Circus, Martin Gardner
Mathematical Puzzles, Geoffrey Mott-Smith