Since it is that festive time of year, we have a fun little puzzle recommended to me by my aunt. Apparently, this is going around via an email chain, stumping people all over the country. Try and then come back here after—but only after—you have figured out the solution, which I provide below.
DO NOT LOOK BELOW THIS LINE!
You peeked, didn’t you? I knew that you would not be able to resist the overwhelming temptation to look below the line before you had worked out a solution. So I didn’t start the answer until later. My prescience is as great as Robin’s!
Now don’t you feel bad? Are you not filled with remorse (he said, deliberately echoing, yet at the same time mangling, a line from A Christmas Story)? Go back and work on it some more until you’ve got it or are truly stumped.
Actually, my purpose is to fill this space with meaningless chatter (I thought of pasting quotes from the New York Times) so that you can’t accidentally see the solution. La, la. How about them Lions? Did they actually win any games this year? I didn’t follow the NFL at all this year. This won’t interest you, but I stand in need of a haircut. I got a haircut once before while I was in Taiwan. Many great things about this country, but barbering Western style hair not one of them. When I removed myself from the barber’s chair, I found myself looking in the mirror at a gentleman who was the spitting image of Moe Howard. Yes, this turned out to be me. Thus, I will wait until I return Stateside before I am shorn.
That enough time wasting? On with the show!
HERE IS THE REAL LINE BELOW WHICH YOU SHOULD NOT PEEK
When most people are confronted with Robin’s regifting problem, they move quickly through it. If they give it any thought, they start with any number and figure the answer they arrive at by the computation can be any number. And that’s a lot of numbers! When the gift grid shows, they immediately hone in on their gift, usually without pausing to pay strict attention to the remainder of the gird.
Then, after locating the prize, they click “Next” and feel shock that Robin guessed correctly. Not content with the apparent psychic powers of a web site, people try again. Another match! What’s going on!
Any two-digit number can be written like this:
a * 10 + b
where a can be any number between 1 and 9, and b can be any number between 0 and 9 (a and b can be equal, of course).
The game asks you to pick a two-digit number and then subtract the first and second digits from it. That is equivalent to this:
answer = a * 10 + b – a – b = a * (10 – 1) = a*9.
Thus, the winning entries—the gifts—can only be at slots 9, 18, 27, …, 81, because a can be 1, 2, …, 9. To preserve symmetry, the game board prints all numbers from 1 to 99; however it is impossible for you to compute an answer larger than 81.
Next time you play, take a look at the gifts at those slots. THEY ARE ALL THE SAME GIFT. So no matter which number you pick, you will be lead to 9, 18, …, 81 and the gifts are always the same. Robin cannot lose.
Each time you play, the gifts are shuffled so that different ones appear at 9, 18, …, 81. Now, most of the numbers between 1 and 99 are, of course, impossible: the mathematical operation given in the rules will never let you deduce any numbers but those divisible by 9. But there are so many numbers displayed (99) that you will not see that many of the gifts repeat. Or if you did, you will tend to think that it does not matter. And gifts are not just repeated at the solution numbers, they are repeated for other numbers, too.
Make sense? Did you get it?