The Game Of Guessing Number

One day I showed Yetao and Ben a book about Recreational Math and Essays, and they immediately loved the book. The book starts with a few number guessing games, which kids wanted to try on me.  As long as you know some basic algebra, most of these games are quite straightforward, for example, one of the game is: Ask the person to choose a number and keep it secret, then do the following operations:

  1. Multiple the number by 5,
  2. Add 6 to the product,
  3. Multiple the sum by 4,
  4. Add 9 to the product,
  5. Multiple the sum by 5.

After these operations, ask to be told the result of the last step, then you can get the original secret number by subtracting 165 from this number and dividing the result by 100.

For Yetao and Ben, this game seems to be magic, and they kept trying it on me, but by using the basic algebra, suppose the original secret number is x, we can easily see:

(1)   \begin{equation*} \begin{split} &(((x \times 5) + 6) \times 4 + 9 ) \times 5 \\ &= ((x \times 5) + 6) \times 20 + 45 \\ &= x \times 100 + 120 + 45 \\ &= x \times 100 + 165 \end{split} \end{equation*}

So if the person tells you his operation result is y, you have the equation

(2)   \begin{equation*} x \times 100 + 165 = y \end{equation*}

Solve it by expressing x using y,  you get:

(3)   \begin{equation*} x = (y- 165) \div 100 \end{equation*}

After kids played this game with me for a few rounds, to enlighten them, I asked them if I can play a game, and they agreed. I asked them to choose a number and kept it secret, and add 1 to it, then tell me the result, then I will know their secret. Yetao and Ben immediately complained that I am cheating, but I do hope one day they can realize the game I suggested is essentially the same with the game they played with me. 🙂