Question 5

Different trick steps lead to a new constant, but the date can still be found.

Step 1 — Understanding the original trick Let $M$ be the month number and $D$ be the day number. The trick gives a final answer. The final answer is $100M + 165 + D$. To find the date, we subtract 165. This leaves us with $100M + D$. For example, if $100M + D = 325$. Then $M=3$ and $D=25$. So, the date is March 25. The number 165 is a constant value. It is added during the trick's steps.

Step 2 — Changing the trick's steps Let us create a new version of the trick. We will change some numbers in the steps. Let's see how this affects the final answer. Here are the new steps we will follow:

1. Take the month number $M$.
2. Multiply it by 5.
3. Add 10 to the result.
4. Multiply this new result by 20.
5. Add the day number $D$.
6. Add 30 to the total.

Step 3 — Calculating the new final answer Let us write down the expression after each step. We start with the month $M$. After step 1, the value is $M$. After step 2, we multiply by 5: $5 \times M$ $= 5M$ After step 3, we add 10: $5M + 10$ After step 4, we multiply by 20: $20 \times (5M + 10)$ $= 20 \times 5M + 20 \times 10$ $= 100M + 200$ After step 5, we add the day $D$: $100M + 200 + D$ After step 6, we add 30: $100M + 200 + D + 30$ $= 100M + 230 + D$

$\boxed{\text{New Final Answer} = 100M + 230 + D}$

Step 4 — Finding the date with the new trick The new final answer is $100M + 230 + D$. Let us call this new final answer $A'$. So, $A' = 100M + 230 + D$. To find $100M + D$, we must subtract 230. $A' - 230$ $= (100M + 230 + D) - 230$ $= 100M + D$

$\boxed{\text{Value to subtract} = 230}$ This value $100M + D$ will give us the month and day. The month $M$ is the number of hundreds. The day $D$ is the remaining number. So, we can still find the original date.

Answer

(i) Yes, we can change the steps in this trick and still find the original date. (ii) For example, we can change the steps to: (1) Take month $M$, (2) Multiply by 5, (3) Add 10, (4) Multiply by 20, (5) Add day $D$, (6) Add 30. (iii) In this new trick, we would have to subtract 230 from the final answer instead of 165 to find the original date.