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Number Guessing Secret

The Secret

The mechanics of this trick are simple; just add the numbers in the upper left corners of the number grids in which the secret number appears. For example, in the 1--125 grids, suppose the secret number is 81. When asked to identify the number grids containing the secret number, the volunteer should select the red, magenta, and black number grids. The numbers in the upper left corners of these grids are 1, 16, and 64, which add up to 81.

The Theory

The theory behind this is as follows. The trick is based on the way that numbers are represented in computers using binary notation. That is, all storage in computers can be viewed as just a sequence of 0s and 1s. Numbers are represented in binary (or base-two) notation which uses just the digits 0 and 1 in a fashion similar to our familiar decimal (or base-ten) notation using digits 0, 1, 2, ..., 9.

The difference between binary and decimal notation is just a matter of place value. In a decimal number like 746, the 6 really does represent 6, but we know that the 4 actually represents 4x10, and the 7 actually represents 7x100. That is, the place value for the rightmost digit is 1, but the next place value to the left is 10, and the next place value is 100. These are the place values in a decimal number because they are successive powers of 10. The rightmost place value is 10^0 = 1, next is 10^1=10, and next is 10^2=100.

In binary notation, the place values are powers of 2 instead of powers of 10. So the place value for the rightmost digit is 1, the place value for the next digit is 2^2 = 4, the next place value is 2^3 = 8, next is 2^4 = 16, then 32, 64, etc. We will not go through a systematic description of how to convert numbers from decimal to binary notation, but you may observe a first several examples:

decimalbinary
10000001
20000010
30000011
40000100
50000101
60000110
70000111
80001000
90001001
100001010

Now, you may understand how the tables are organized in the number-guessing trick. The first table contains all those numbers that have a 1 in the 1s place when written in binary. The second table contains all those numbers that have a 1 in the 2s place when written in binary. The next table contains all those numbers that have a 1 in the 4s place when written in binary. Thus, by know which tables the secret number is contained in, we know which place values need to be added up to obtain the value of the secret number. And in each table, the relevant place value is found in the upper left corner. That's why the "mechanics" described above will work.

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