Contest #1
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A day in Mathematical Thinking and Creative Problem Solving
Problem 1: A game of tokens
There is a stack of 25 tokens and 2 players. The players take turns one by one. In their turn, a player can pick either 1 or 2 tokens. They continue until all the tokens are finished.
Winner : The one who picks the last token is the winner.
While Ishaan and Nidhi were playing this game, Nidhi picked the 22nd token and said, ‘I am winning this for sure’. How was she confident of winning?
1a Response to question 1a
And she did win. Next game, Ishaan thought a lot about how he could win. He finally came up with a strategy, and claimed that no could ever beat him if he could start the game, i.e. have the first turn. What could be his strategy?
1b
Problem 2: The Travelling Bee
The initial position of a Bee is shown in the Beehive.
It can move in only one general direction, namely, to the right. For example, if the bee was in cell H, it could move only to I or J from there.
In how many ways can the bee reach the cell ‘D’ from the starting point?
2a
In how many ways can the bee reach the the cell ‘H’ from the starting point?
2b 13 22 34 55
How did you find it?
Problem 3: The Secret Code
Adapted from Dan Finkel’s Ted Talk: 5 principles of Extraordinary Math teaching
An intelligence agency has found a secret code in which there is a strange way of representing numbers with colored circles. They have lost the 2nd part of the code, which had numbers upto 100.
What are some patterns that you observe here?
Patterns
The agency needs the color coding of the numbers 64 and 75. Which colors would be present in 64 and 75? How would the colors be split?
64
75
Problem 4: The Squareable Numbers
A number n is called squareable if a square can be made out of precisely n squares. For example, 4 is squareable: 4 small squares can form a square.
11 is squareable: 11 different squares can be fit together to form another square.
Is 10 a squareable number? Why?
4a
Which of these are squareable numbers?
4b 3 12 9 14
Are all even numbers squareable?
4c Yes No
Why do you think so?
Someone claimed that all numbers in this series (consecutively adding 3) {1,4,7,10,……} are definitely squareable. What could be a possible explanation for this claim?
4d
Problem 5: Can I buy the painting?
30 people came to buy an art piece.
Each of them has coins of a different denomination. The 1st person has unlimited coins of Re 1, 2nd person has unlimited coins of Rs 2, and so on. The 30th person has unlimited coins of Rs. 30.
The price for the art piece was to be paid in the exact amount, neither a penny less nor more.
Each of them started to count their money. Each of them could pay the exact amount, except for 2 consecutive buyers.
Which were these 2 buyers? How do you know?
5a
Problem 6: Let’s Play Bingo
The game: Each player gets a 4×4 grid ticket with a number in each square. A pair of dice is rolled. The product of the numbers on the dice is announced. As a number is announced, the players can cut it off.
The first player to get all the numbers cut on the ticket wins. Below are the 3 available tickets.
Which of these ticket should you pick for the maximum chances of winning? Why?
6a
Which of these has the lowest chances of winning? Why?
6b
Problem 7: Circles and Fractions
Can you describe what is happening in the pictures above as we move from one picture to the next?
7a
What is the fraction of black color in the 2nd and 3rd image?
7b
What would be the fraction of black color in the 8th image? How are you thinking about this? Do you see a pattern?
7c
Problem 8: The Mystery of odd numbers
A mathematician, while playing with numbers, observed something very interesting about odd numbers. She observed the following :
After trying out with a few more numbers and playing around, she claimed the following.
Why is this pattern emerging? Will this be always true? Why?
If not, can you think of a counter-example?
8a
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