Algebra Play | IT

Question 6

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

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Solution
Understand the Question
  • A 'Think of a Number' trick involves performing a series of mathematical operations on an unknown starting number such that the original number cancels out, leaving a constant final result.
  • By representing the unknown number algebraically with a variable (e.g., xx), we can create and prove that the result is independent of the number chosen.

Step 1 · Define the Trick's Instructions

  1. Think of any number.
  2. Multiply it by 22.
  3. Add 66 to the result.
  4. Divide the new number by 22.
  5. Subtract the original number you thought of.

Step 2 · Prove the Trick Algebraically

Let the chosen number be xx.

  • Think of a number: xx

  • Multiply by 22:

x×2=2x\begin{aligned} x \times 2 &= 2x \end{aligned}

  • Add 66: 2x+62x + 6

  • Divide by 22:

2x+62=2x2+62=x+3\begin{aligned} \dfrac{2x + 6}{2} &= \dfrac{2x}{2} + \dfrac{6}{2} \\[0.6em] &= x + 3 \end{aligned}
  • Subtract the original number (xx):
(x+3)−x=x+3−x=3\begin{aligned} (x + 3) - x &= x + 3 - x \\ &= 3 \end{aligned}
Answer

The final answer to the trick is always 33.

Common Mistakes
  • Division Distribution Error: When dividing 2x+62x + 6 by 22, forgetting to divide both terms by 22, incorrectly writing 2x+62=x+6\dfrac{2x + 6}{2} = x + 6.
  • Incomplete Cancellation: Forgetting to subtract the original number at the final step, which would leave the answer dependent on xx rather than a constant trick value.

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 22).

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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