Algebra Play | IT

Question 5

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.

Solution

Understand the Question

The original trick results in the expression $100M + 165 + D$ , where $M$ is the month and $D$ is the day.
Subtracting the constant $165$ isolates $100M + D$ , where the hundreds place gives the month $M$ and the remaining digits give the day $D$ .
By altering the operations, we can arrive at a new expression of the form $100M + C + D$ , where $C$ is a different constant. Subtracting $C$ allows us to recover the original date in the exact same manner.

Step 1 · Formulate New Trick Steps and Calculate the Expression

Consider a new set of operations starting with month number $M$ and day number $D$ : Diagram 1

Multiply the month $M$ by $5$ : $5 \times M = 5M$
Add $10$ : $5M + 10$
Multiply by $20$ :

\begin{aligned} 20 \times (5M + 10) &= 20 \times 5M + 20 \times 10 \\ &= 100M + 200 \end{aligned}

Add the day $D$ : $100M + 200 + D$
Add $30$ : $100M + 200 + D + 30 = 100M + 230 + D$

Step 2 · Recover the Original Date

Let the new final result be $A' = 100M + 230 + D$ .

Subtract the new constant $230$ :

\begin{aligned} A' - 230 &= (100M + 230 + D) - 230 \\ &= 100M + D \end{aligned}

The hundreds digit gives the month $M$ and the last two digits give the day $D$ . Hence, the date can still be determined.

Answer

Yes, we can modify the steps. In this example, subtracting $230$ from the final answer instead of $165$ gives $100M + D$ , revealing the original date.

Common Mistakes

Coefficient Error on $M$ : Failing to ensure the coefficient of $M$ multiplies out to exactly $100$ . If the coefficient is not $100$ , the month and day digits will overlap and cannot be easily separated.
Subtracting the Wrong Constant: Forgetting to account for how earlier additions get scaled by later multiplications when tracking the constant term (e.g., adding $10$ and then multiplying by $20$ contributes $200$ , not $10$ ).

More questions in IT

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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.

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Let us start with the simplest pyramid.

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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

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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.

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Q14

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 $a > b$ ?