Question 7

Context: Think of a number trick:

1. Think of a number.
2. Double it.
3. Add four.
4. Divide by two.
5. Subtract the original number you thought of.

Q. I predict you get 2. Am I right? Try it out with different starting numbers. Do you always end up with the same value, 2? Why?

Context: Consider the following number trick:

1. Think of a number.
2. Double it.
3. Add four.
4. Divide by two.
5. Subtract the original number you thought of. (This trick always results in 2).

Q. How would you change this game to make the final answer 3? What about 5?

Context: Consider the following number trick:

1. Think of a number.
2. Double it.
3. Add four.
4. Divide by two.
5. Subtract the original number you thought of. (This trick always results in $2$).

Q. Can you come up with more complicated steps that always lead to the same final value?

Context: In the date trick, the final answer is given by $100M + 165 + D$, where $M$ is the month and $D$ is the day.

Q. Find the dates if the final answers are the following:

(i) 1269

(ii) 394

(iii) 296

Context: In the date trick, the final answer is given by $100M + 165 + D$, where $M$ is the month and $D$ is the day.

Q. Can you change the steps in this trick and still find the original date? Instead of subtracting 165 from the final answer, you might have to subtract some other number.

Try to devise your own 'Think of a Number' trick.

Fill the following pyramids:

What is the relationship between the numbers in the bottom row and the number at the top?

Let us start with the simplest pyramid.

What about a pyramid with three rows?

Using letter numbers for the bottom row, we can write an expression for the top row.

In the following grids, find the values of the shapes and fill in the empty squares:

Context: 6.5 The Largest Product

Q. Fill the digits 2, 3, and 5 in $\square\square \times \square$, using each digit once. What is the largest product possible?

Context: Mukta's trick: Choose a 2-digit number of different digits, reverse the digits to get another number, find their difference, and divide the result by 9.

Q. If we choose other 2-digit numbers, and follow the steps, will there always be no remainder?

Context: Suppose a two-digit number is $ab$. When it is reversed, the new number is $ba$. If $b > a$, the difference is $ba - ab = 9(b - a)$, which is divisible by 9.

Q. Can you work out what happens if $a > b$?