Secret of the Seven. Effective trick and its solution

EFFECTIVE FOCUSES AND THEIR CLUES
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Secret of the Seven. Focus Secret

Spectacular tricks and their clues

Directory / Spectacular tricks and their clues

Focus Description:

All "mysterious" nine properties are explained by the simple fact that this figure is the last in the decimal number system we use. In the octal number system, the seven has the same curious properties.

This assertion is easy to verify. First of all, let's make a list of sixteen numbers, denoting them in the octal system, and write down their equivalents in the decimal system next to each other.

octal system Decimal system

1 1

2 2

3 3

4 4

5 5

6 6

7 7

10 8

11 9

12 10

13 11

14 12

15 13

16 14

17 15

20 16

Suppose we took the number 341 (an octal notation) and subtracted the number 143 from it, obtained by reversing the order of the numbers. First, subtract 3 from 11. In decimal, this would mean the same as subtracting 3 from 9. The answer would be 6. But the number 6 in both number systems means the same number, so the difference between 11 (recording in octal ) and 3 is equal to 6. Continuing further subtraction in the same way, we get 176 in the answer (record in octal system):

341
-
143
---
176

You notice that the digit in the middle is a seven and that the sum of the extreme digits is also seven. Exactly the same thing happens here as in the decimal version of this trick we described earlier, except that the key number is seven, not nine.

All other tricks based on the properties of the nine in the decimal system can be subjected to a similar test. Moreover, for each of them there is a corresponding focus in the octal system, but the role of the "mysterious number" will belong to the seven. By choosing the appropriate number system, you can transfer special properties to any desired number. Thus, it becomes obvious that these properties do not stem from the intrinsic features of nine, but only from the fact that it is the last digit in our decimal number system.

It is a common mistake to confuse the intrinsic properties of a number with properties derived from its location in a given number system. So, at one time it was thought that for some hidden reasons, among the numbers depicting an infinite non-periodic decimal fraction denoting the number i, the seven is found on average less often than other numbers. “There is only one number so unequal among other numbers that it is unbelievable that it could be an accident,” wrote Dr. Augustus de Morgan, “and this number is the mysterious seven,” De Morgan wrote this, of course, not seriously; he knew well that the digits of the number i in a different number system would be completely different.

In fact, even in the decimal system, the apparent rarity of the occurrence of a seven in the number i is due to an error made by William Shanks in calculating this number. Back in 1873, after fifteen years of hard work, Shanks calculated the number π with seven hundred and seven decimal places (the mistake he made at the 528th sign nullified all subsequent calculations). In 1949, the ENIAC computer, so to speak, in the form of a break from more complex tasks, I calculated with more than 2000 correct decimal places. At the same time, no "mysterious" deviations in the frequency of occurrence of any number were found.

It is proved that the frequency of occurrence of any digit in the decimal expansion of almost all numbers is the same and equals - 1/10 (and in the expansion with base m it is 1/m). Numbers for which this does not hold are said to form a set of measure zero. i.e., they can be enclosed in a system of numerical intervals with any small total length.

Author: M.Gardner

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