A "magic" trick: think of any number, double it, add 10, halve it, then subtract the starting…
A "magic" trick: think of any number, double it, add 10, halve it, then subtract the starting number — the answer is always 5. Kartik thinks this only works for small numbers. His sister says it works for every number, however large. Who is right?
Who do you think is right? Pick one.
The answer: His sister is right, it works for every number
Why
Kartik's suspicion is a healthy instinct — a trick that only seems to work in a few tried cases can very often break down somewhere else. This particular trick can actually be checked completely, for every possible number at once, by using a letter to stand for 'any number': call the starting number x. Doubling gives 2x; adding 10 gives 2x+10; halving gives (2x+10)÷2, which is x+5; subtracting the original x gives x+5−x, which is just 5 — the x's cancel out completely, no matter what number x actually was. Test it: try it with a huge number, like x=999 — double is 1998, add 10 is 2008, halve is 1004, subtract 999 is 5 — still 5, exactly as the algebra predicted. His sister is right: because the starting number cancels out in the working, this trick gives 5 for every number, not just small ones — that is exactly what makes it a genuine rule and not a coincidence noticed in a few examples.
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