Q1.
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?
Answer
Yes — the answer is 2 every single time. Try three very different starting numbers:
| Step | Start with 7 | Start with 40 | Start with 1/2 |
|---|---|---|---|
| Think of a number | 7 | 40 | 1/2 |
| Double it | 14 | 80 | 1 |
| Add four | 18 | 84 | 5 |
| Divide by two | 9 | 42 | 5/2 |
| Subtract the original number | 2 | 2 | 2 |
Trying numbers shows that it works. To see why, do the same five steps to a letter-number x:
Think of a number: x
Double it: 2x
Add four: 2x + 4
Divide by two: x + 2
Subtract the original number: (x + 2) – x = 2
Double it: 2x
Add four: 2x + 4
Divide by two: x + 2
Subtract the original number: (x + 2) – x = 2
Why it happens: the letter-number x is not one particular number — it stands for whatever you thought of. So this single line of working is a check on every starting number at the same time. Doubling makes the amount of x equal to 2x; halving after adding 4 brings it back to one x and turns the 4 into 2. The last step removes that one x, and only the 2 is left. Nothing about x survives to the end, so nothing about the answer can depend on x.
Tip: notice why step 3 must come between doubling and halving. The 4 is added when there are two x’s, so halving splits the 4 as well and leaves 2. Add the 4 after halving and the trick would end at 4 instead.