Algebra Play | IT

Question 3

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?

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Solution

Algebra helps us show a number trick always gives the same result.

Step 1 — Understanding the original trick

Let us use a variable to represent the number. Let xx be the number you think of. We will follow the steps of the original trick.

First, we double the number. 2x2x Next, we add four to this result. 2x+42x + 4 Then, we divide the entire expression by two. 2x+42\frac{2x + 4}{2} =2x2+42= \frac{2x}{2} + \frac{4}{2} =x+2= x + 2 Finally, we subtract the original number xx. (x+2)−x(x + 2) - x =2= 2

2\boxed{2}

Step 2 — Designing new complicated steps

We want to create a new trick. It must also always result in 2. Let us make the steps more complex. Here are the new steps we designed:

  1. Think of a number.
  2. Multiply it by six.
  3. Add twelve to the result.
  4. Subtract two times your original number.
  5. Divide the new result by four.
  6. Subtract one from this value.
  7. Subtract the original number you thought of.

Step 3 — Verifying the new trick

Let us check if our new trick works. Let xx be the number you think of. We will follow the new steps using xx.

First, we multiply the number by six. 6x6x Next, we add twelve to this result. 6x+126x + 12 Then, we subtract two times the original number xx. (6x+12)−2x(6x + 12) - 2x =4x+12= 4x + 12 After that, we divide the new result by four. 4x+124\frac{4x + 12}{4} =4x4+124= \frac{4x}{4} + \frac{12}{4} =x+3= x + 3 Next, we subtract one from this value. (x+3)−1(x + 3) - 1 =x+2= x + 2 Finally, we subtract the original number xx. (x+2)−x(x + 2) - x =2= 2

2\boxed{2}

Answer

(i) Here are more complicated steps that always lead to 2:

  1. Think of a number.
  2. Multiply it by six.
  3. Add twelve to the result.
  4. Subtract two times your original number.
  5. Divide the new result by four.
  6. Subtract one from this value.
  7. Subtract the original number you thought of.

More questions in IT

Q1

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?

Q2

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?

Q3

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?

Q4

Context: In the date trick, the final answer is given by 100M+165+D100M + 165 + D, where MM is the month and DD is the day.

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

(i) 1269 (ii) 394 (iii) 296

Q5

Context: In the date trick, the final answer is given by 100M+165+D100M + 165 + D, where MM is the month and DD 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.

Q6

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

Q7

Use the same rule to fill these pyramids:

Q8

Fill the following pyramids:

Q9

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

Let us start with the simplest pyramid.

Q10

What about a pyramid with three rows?

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

Q11

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

Q12

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?

Q13

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?

Q14

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

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

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