Object and number trick

Question

You are telling your friends Adam and Steve a trick and here is how it goes:

Tell Adam to take out $3$ objects from his pocket which all the objects have a different amount of letters. Lets say Adam picks keys, watch, bag. Now ask Adam to think of one of the objects he picked. (Lets say Adam picked watch). Now ask Adam to take the number of letters in the object (which is $5$) and do the following, multiply the number by $5$, add $3$ and then double the total. Now have steve whisper in adam's hear a number from $1-9$, whatever number steve whispered to adam must be added to adam's total.

you are able to tell what object adam was thinking about and what number steve told adam, how?

Like i said:

Lets say adam was thinking about a watch which has $5$ letters. Now:

$$2[(5 \times 5)+3]=56$$

Lets say Steve whispered $7$, then $56+7=63$

I did this a couple of more times with different numbers but wasnt able to see the pattern. does anyone else see it?

Answer 1

It should help to write out algebraically what exactly is happening. Suppose that the number of letters for Adam's object is $x$. Adam will then quintuple it to get $5x$ and add three to get $5x+3$. He then doubles it to get $10x+6$. Then, another number $k$ is added to it to get $10x+6+k$. Can you see how to recover $x$ and $k$ from this? Try playing around with it and maybe making a table of the value of $10x+6+k$ depending on $x$ and $k$.

(More simpliy: Suppose you knew $10x+k$? Could you figure it out then? Try plugging suitable values for $x$ and $k$ into this and see a pattern)

Answer 2

If $x$ is the number of letters in Adam's word and Steve's number is $n$ then the number you get is $2(5x+3)+n=10x+6+n$.

To get Steve's number you subtract $6$ from the total and take the remainder from dividing by $10$. Subtracting that number from the total gives you $10x+6$ so you subtract $6$ from that number and divide by 10 to get the number of letters in Adam's word.