Algebra Play | FIO

Question 19

Karim and the Genie

Karim was taking a nap under a tree. He had a dream about a magical lamp and a genie. He heard a voice saying, “I have come to serve you, Oh master”. He woke up and to his surprise, it was a genie!

“Do you want to make money?”, asked the genie. Karim nodded dumbly in bewilderment. The genie continued, “Do you see the banyan tree over there? All you have to do is go around it once. The money in your pocket will double”.

Karim immediately started towards the tree, only to be stopped by the genie. “One moment!”, said the genie. “Since I am bringing you great riches, you should share some of your gains with me. You must give me 8 coins each time you go around the tree.”

Thinking that was a trifling amount, Karim readily agreed.

He went around the tree once. Just as the genie had said, the number of coins in his pocket doubled! He gave 8 coins to the genie. He made another round. Again the number of coins doubled. He gave 8 more coins to the genie. He went around the tree for the third time. The number of coins doubled again, but to his horror, he was left with only 8 coins, exactly the number of coins he owed the genie!

As Karim began to wonder how the genie tricked him, the genie let out a loud laugh and disappeared.

(i) How many coins did Karim initially have?

(ii) For what cost per round should Karim agree to the deal, if he wants to increase the number of coins he has?

(iii) Through its magical powers, the genie knows the number of coins that Karim has. How should the genie set the cost per round so that it gets all of Karim's coins?

Solution

We can use algebra to track Karim's coins through each round and determine the initial amount, as well as conditions for gaining or losing coins.

Step 1 — Initial number of coins

Let $N_0$ be the number of coins Karim initially had. Let $N_i$ be the number of coins Karim had after the $i$ -th round and after paying the genie. The genie charges 8 coins per round.

After the first round, Karim's coins double, then he pays 8 coins. $N_1 = 2N_0 - 8$

After the second round, his current coins double, then he pays 8 coins. $N_2 = 2N_1 - 8$

After the third round, his current coins double, then he pays 8 coins. $N_3 = 2N_2 - 8$

The problem states that after the third round, Karim was left with exactly 8 coins. So, we know $N_3 = 8$ . We can substitute this value and work backwards to find $N_0$ .

First, we find $N_2$ : $8 = 2N_2 - 8$ $8 + 8 = 2N_2$ $16 = 2N_2$ $N_2 = \frac{16}{2}$

$\boxed{N_2 = 8 \text{ coins}}$

Next, we find $N_1$ : $N_2 = 2N_1 - 8$ $8 = 2N_1 - 8$ $8 + 8 = 2N_1$ $16 = 2N_1$ $N_1 = \frac{16}{2}$

$\boxed{N_1 = 8 \text{ coins}}$

Finally, we find $N_0$ : $N_1 = 2N_0 - 8$ $8 = 2N_0 - 8$ $8 + 8 = 2N_0$ $16 = 2N_0$ $N_0 = \frac{16}{2}$

$\boxed{N_0 = 8 \text{ coins}}$

Diagram 1

Step 2 — Condition for increasing coins

Let $N$ be the number of coins Karim has at the start of any round. Let $k$ be the cost per round that Karim pays to the genie. After going around the tree, Karim's coins double to $2N$ . Then he pays $k$ coins to the genie, so he is left with $2N - k$ coins. For Karim to increase the number of coins he has, the amount after the round must be greater than the amount before the round.

So, we set up an inequality: $2N - k > N$

To find the condition for $k$ , we can subtract $N$ from both sides. $N - k > 0$ $N > k$

This means the cost per round ( $k$ ) must be less than the number of coins Karim has at the start of that round.

Step 3 — Genie getting all coins

Let $N$ be the number of coins Karim has at the start of a round. Let $k$ be the cost per round that the genie sets. The genie wants to get all of Karim's coins. This means Karim should be left with 0 coins after paying the genie. After doubling, Karim has $2N$ coins. If he is left with 0 coins after paying $k$ , then: $2N - k = 0$

To find the cost $k$ , we can add $k$ to both sides. $2N = k$

So, the genie should set the cost per round to be twice the number of coins Karim has at the start of that round.

Answer

(i) Karim initially had 8 coins. (ii) The cost per round should be less than the number of coins Karim has at the start of that round. (iii) The genie should set the cost per round to be twice the number of coins Karim has at the start of that round.

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If the bottom row of an $n$ row pyramid contains the first $n$ Virahāṅka-Fibonacci numbers, what can we say about the numbers in the pyramid? What can we say about the number at the top?

Fill the digits 1, 3, and 7 in $\square\square \times \square$ to make the largest product possible.

Fill the digits 3, 5, and 9 in $\square\square \times \square$ to make the largest product possible.

In the trick given above, what is the quotient when you divide by 9? Is there a relationship between the two numbers and the quotient?

Q10

In the trick given above, instead of finding the difference of the two 2-digit numbers, find their sum. What will happen? For example:

We start with 31. After reversing we get 13. Adding 31 and 13, we get 44.
We start with 28. After reversing we get 82. Adding 28 and 82, we get 110.
We start with 12. After reversing we get 21. Adding 12 and 21, we get 33.

Observe that all these numbers are divisible by 11. Is this always true? Can we justify this claim using algebra?

Q11

Consider any 3-digit number, say $abc$ ( $100a + 10b + c$ ). Make two other 3-digit numbers from these digits by cycling these digits around, yielding $bca$ and $cab$ . Now add the three numbers. Using algebra, justify that the sum is always divisible by 37. Will it also always be divisible by 3? [Hint: Look at some multiples of 37.]

Q12

Consider any 3-digit number, say $abc$ . Make it a 6-digit number by repeating the digits, that is $abcabc$ . Divide this number by 7, then by 11, and finally by 13. What do you get? Try this with other numbers. Figure out why it works. [Hint: Multiply 7, 11 and 13.]

Q13

There are 3 shrines, each with a magical pond in the front. If anyone dips flowers into these magical ponds, the number of flowers doubles. A person has some flowers. He dips them all in the first pond and then places some flowers in shrine 1. Next, he dips the remaining flowers in the second pond and places some flowers in shrine 2. Finally, he dips the remaining flowers in the third pond and then places them all in shrine 3. If he placed an equal number of flowers in each shrine, how many flowers did he start with? How many flowers did he place in each shrine?

Q14

A farm has some horses and hens. The total number of heads of these animals is 55 and the total number of legs is 150. How many horses and how many hens are on the farm?

Can you solve this without letter-numbers?

[Hint: If all the 55 animals were hens, then how many legs would there be? Using the difference between this number and 150, can you find the number of horses?]

Q15

A mother is 5 times her daughter's age. In 6 years' time, the mother will be 3 times her daughter's age. How old is the daughter now?

Q16

Two friends, Gauri and Naina, are cowherds. One day, they pass each other on the road with their cows. Gauri says to Naina, "You have twice as many cows as I do". Naina says, "That's true, but if I gave you three of my cows, we would each have the same number of cows". How many cows do Gauri and Naina have?

Q17

I run a small dosa cart and my expenses are as follows:

Rent for the dosa cart is ₹5000 per day.
The cost of making one dosa (including all the ingredients and fuel) is ₹10.

(i) If I can sell 100 dosas a day, what should be the selling price of my dosa to make a profit of ₹2000?

(ii) If my customers are willing to pay only ₹50 for a dosa, how many dosas should I aim to sell in a day to make a profit of ₹2000?

Q18

Evaluate the following sequence of fractions:

$\frac{1}{3}, \frac{(1 + 3)}{(5 + 7)}, \frac{(1 + 3 + 5)}{(7 + 9 + 11)}$

What do you observe? Can you explain why this happens?

[Hint: Recall what you know about the sum of the first $n$ odd numbers.]

Q19