Games and Winning Strategies

Numbers can also be used to play games and develop winning strategies.

Here is a famous game called 21. Play it with a classmate. Then try it at home with your family!

Rules for Game #1 :

The first player says 1, 2 or 3. Then the two players take turns adding 1, 2, or 3 to the previous number said. The first player to reach 21 wins!

Play this game several times with your classmate. Are you starting to see the winning strategy?

Which player can always win if they play correctly? What is the pattern of numbers that the winning player should say?

There are many variations of this game. Here is another common variation:

Rules for Game #2: The first player says a number between 1 and 10. Then the two players take turns adding a number between 1 and 10 to the previous number said. The first player to reach 99 wins!

Play this game several times with your classmate. See if you can figure out the corresponding winning strategy in this case! Which

player can always win? What is the pattern of numbers that the winning player should say this time?

Make your own variations of this game — decide how much one can add at each turn, and what number is the winning number. Then play your game several times, and figure out the winning strategy and which player can always win!

Figure it Out 8

1. There is only one supercell(number greater than all its neighbours 🟩) in this grid. If you exchange two digits of one of the numbers, there will be 4 supercells. Figure out which digits to swap.

Solution :

2. How many rounds does your year of birth take to reach the Kaprekar constant?

If your year of birth is 2000

Step 1: Now from digits of number 2000

Here largest number = 2000 and smallest number =

Let’s subtract them = 2000 – 0002 =

Step 2: Now from digits of number 1998

Here largest number = 9981 and smallest number =

Let’s subtract them = 9981 – 1899 =

Step 3: Now from digits of number 8082

Here largest number = 8820 and smallest number =

Let’s subtract them = 8820 – 0288 =

Step 4: Now from digits of number 8532

Here largest number = 8532 and smallest number =

Let’s subtract them = 8532 – 2358 =

which is a Kaprekar constantvariable.

Hence it took rounds to reach the Kaprekar constant from 2000

3. We are the group of 5-digit numbers between 35,000 and 75,000 such that all of our digits are odd. Who is the largest number in our group? Who is the smallest number in our group? Who among us is the closest to 50,000?

The largest number with all odd digits (different) =

The largest number with all odd digits (repetitive) =

The smallest number (non repetitive) =

The smallest number (repetitive) =

Closest to 50,000 (in case of non-repetition) =

Closest to 50,000 (in case of repetition) =

4. Estimate the number of holidays you get in a year including weekends, festivals and vacation. Then, try to get an exact number and see how close your estimate is.

Estimation

Instruction

Actual

5. Estimate the number of liters a mug, a bucket and an overhead tank can hold.

1. Mug

A standard mug usually holds about 250–3502000-3000 milliliters (ml).

Estimate: liters

2. Bucket

A typical household bucket holds about 10–20100-150 liters.

Estimate: liters

3. Overhead Tank

verhead water tanks vary widely depending on usage: For homes: 500–200010,000-20,000 liters is common.

Estimate: 1000100 liters

6. Write one 5-digit number and two 3-digit numbers such that their sum is 18,670.

Solution:

5 digit number = 1 8 0 0 0

3 digit number = 6 7 0

Sum = 1 8 000 + 670 =

7. Choose a number between 210 and 390. Create a number pattern similar to those shown in Section 3.9 that will sum up to this number.

Solution:

Sum of No. = 5 × 1 =

(+) 10 × 3 =

(+) 15 × 5 =

(+) 20 × 7 = 140 =

which lies between 210150 and 390350.

8. Recall the sequence of Powers of 2 from Chapter 1, Table 1. Why is the Collatz conjecture correct for all the starting numbers in this sequence?

Solution:

The square of power of 2 is : 1,2,4, 8, 16, 32, 64

Let’s take the number’ 64 as per Collatz Conjecture

64 is even, divide by 2 =

32 is even, divide by 2 =

16 is even, divide by 2 =

8 is even, divide by 2 =

4 is even, divide by 2 =

2 is even, divide by 2 =

Hence Collatz conjecture is correct in all numbers in the power of sequence.

As it is power of 2, and in Collatz Conjecture even number is divided by 2 in each step.

9. Check if the Collatz Conjecture holds for the starting number 100.

As per Collatz conjecture rule: Starts with any nymber; if the number is even, take half of it; if the number is odd, then multiply it by 3 and add 1; and repeat.

The sequence formed with starting number 100 is as follows: 100,,,76,38,19,58,29,,44,22,11,34,17,52,26,13,,20,10,5,,8,4,2,1.