The Magic Numbered Boards

My second grade teacher used to have a magic trick. She had a series of boards with a bunch of numbers written on it, usually from 1 to 40 or so.

The kids had to choose a number while the teacher wasn’t looking (she waited out in the hallway). She would then come back and show the class each board and ask them if the number was on it. Yes, yes, no, no, yes… After we had gone through all the boards, she would tell us the number. Magic!

For the second grader that I was, it was pretty impressive. After the five boards or so, she knew. She must have had a damn good memory!

But I suspected something was off. There had to be a trick, of course. The whole group thought she must have been reading the erasures of the black board we were writing the number on, or looking through the door window. And they made damn sure she didn’t. But this wasn’t it.

Time after time, I observed. And figured it out. When I asked to try at her place, I managed the same feat. The class was pretty stunned.

The Experiment

Not counting computer science and math majors, can a college-educated adult, with or without a science background, crack the puzzle?

I’ve fashioned the following “boards” on scraps of paper:

+-------------------+ +-------------------+ +-------------------+
| Board  2          | | Board  1          | | Board  8          |
+-------------------+ +-------------------+ +-------------------+
|  2 14 15 11 18 19 | |  1  3 11 13 19 27 | |  8 10 12 14 15 26 |
|  3  6  7 10 22 23 | | 15  5  9 21 33 29 | |  9 11 13 24 28 25 |
| 26 27 30 31 34 35 | | 17  7 25 23 31 35 | | 30 31 40 41 27 29 |
| 38 39 42 43 46 47 | |    47 39 53 45 37 | | 44 45 43 42 47 46 |
| 50 51 54 55 58 59 | |       41 43 49 51 | |       56 57 59 63 |
| 62 63             | |    63 55 61 57 59 | |       58 60 61 62 |
+-------------------+ +-------------------+ +-------------------+

+-------------------+ +-------------------+ +-------------------+
| Board 32          | | Board  4          | | Board 16          |
+-------------------+ +-------------------+ +-------------------+
| 32 33 39 40 47 46 | |  4  5 12 13 20 21 | | 16 17 20 21 24 25 |
| 37 34 38 41 48 45 | |  6  7 14 15 22 23 | | 18 19 22 23 26 27 |
| 36 35 49 42 43 44 | | 28 29 36 37 44 45 | | 28 29 48 49 52 53 |
| 52 51 50 56 57 58 | | 30 31 38 39 46 47 | | 30 31 50 51 54 55 |
| 53 54 55 61 60 59 | | 52 53 60 61       | | 56 57 60 61       |
|          62 63    | | 54 55 62 63       | | 58 59 62 63       |
+-------------------+ +-------------------+ +-------------------+

and had my housemates choose a number between 1 and 60. Then show them each card successively and ask them if their number was there. Half a second after we were done through the six cards, I would tell them their number.

Can you?

I made a point of hiding the cards before and after the trick, while we were discussing, and to distract them a bit with some talking at some points.

We discussed the trick afterwards, and in between runs.

Some of their observations:

Test Subjects A and B: Mathieu and Gabrielle

Mathieu and Gabrielle are studying psychology at Université de Montréal. They studied social science in Cegep.

• Does this have something to do with the thing you do with your hand? (I count to 31 on my hand, sometimes)
• 63 appears on all of the cards

Test Subject C: Iké

Iké is studying environment at McGill University. He studied Pure Science in Cegep.

• To each card I say yes or no, so it’s kind of like a 1 or 0 on each card. It’s kind of a binary thing.

The Explanation

Coming back to second grade, I noticed a strange thing: the boards aren’t numbered sequentially. They are numbered 1, 2, 4, 8, 16, 32. What a strange sequence for numbering boards. Something’s up.

There was still a mystery to me: how did she make those boards? Because they were clearly handmade.

What if somehow, those numbers meant something? What if they… added up to the magical number? Correct.

My teacher was using the same logic as the one behind the binary numeral system. Either the number is one a board, or it is not. With each successive board, we go bit by bit (literally bits). Each board represents a binary digit (bit!) in the number. Over six boards, we could select one number out of 64 (or 2^6 ). So the number 37, 100101 in binary, the sum of positions 32, 4 and 1, would appear on boards 32, 4 and 1!

In second grade, it was magic. Well, math-magic. It wasn’t until I was much (much!) older, that I uncovered the link with combinatorics and binary, and thus solved the real mystery. I interested myself to binary in a research project of our choice in secondary One (age 13), where I became acquainted with it for the first time. Had it not been for that, the educational system would have teached me those in Cegep (age 18), as part of my post-secondary Pure Science program.