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This trick is extremely similar to the Magic Boxes, except using ternary, also known as "base three". As you might have noticed, the numbers in the rows all contain a specific pattern. Every time you click a button, though, new sets of numbers appear.
The trick is that every set of numbers contains a common ternary digit. The each group of rows solves a single ternary digit. The first group solves the last digit. Each row is of the format ??x, where ? is any number and x is the common ternary digit. One row is of the form ??0, another ??1, and the last ??2. When you click the button next to the row that contains your number, the computer stores the last ternary digit.
The next group that appears are of the format ?x?. That is, the rows are ?0?, ?1?, and ?2?. Again, clicking the button stores the ternary digit.
The last group of numbers contain digits of the form x??. The rows, therefore, are 0??, 1??, and 2??
When you have clicked the last button, the computer now has the three ternary digits of your number. To convert this number into decimal is a trivial task.
Please read Magic Boxes: How does this work? to get a fundamental grasp of binary before proceeding.
The only difference between binary and ternary is the "10 substitution." In binary, we substituted the number 2 for the number 10. In ternary, the substituted number is 3.
For example, here's a sample ternary conversion to base 10.
210
=(2 x 100) + (1 x 10)
=(2 x 10²) + (1 x 10)[Here we substitute 3 for 10]
=(2 x 3²) + (1 x 3)
=(2 x 9) + 3
= 18 + 3
= 21
121
=(1 x 100) + (2 x 10) + (1 x 1)
=(1 x 10²) + (2 x 10) + 1[Substituting]
=(1 x 3²) + (2 x 3) + 1
=(1 x 9) + 6 + 1
= 9 + 7
= 16
There is one other aspect that separates this trick from the magic boxes. To better conceal the patterns formed by the numbers, the numbers in the rows are scrambled, and the rows themselves are scrambled. To do this trick on your own, you should probably organize the numbers and rows. For your convenience, here they are:
??0 | ??1 | ??2 |
---|---|---|
0 3 6 9 12 15 18 21 24 | 1 4 7 10 13 16 19 22 25 | 2 5 8 11 14 17 20 23 26 |
?0? | ?1? | ?2? |
0 1 2 09 10 11 18 19 20 | 3 4 5 12 13 14 21 22 23 | 6 7 8 15 16 17 24 25 26 |
0?? | 1?? | 2?? |
0 1 2 3 4 5 6 7 8 | 9 10 11 12 13 14 15 16 17 | 18 19 20 21 22 23 24 25 26 |
Also, similar to the Magic Boxes, you can calculate the number by adding up the first numbers of every row the number is in. This only works if the numbers are sorted!