“7”: Seven Wonders, Seven Colors of a Rainbow
“Seven” 切换中文版： “7”： 七色彩虹，七色桥
> A rainbow has 7 colours.
To be precise, rainbows span a continuous spectrum of colours. Any distinct bands perceived are an artefact of human colour vision. The most commonly cited and remembered sequence of the rainbow is Newton’s sevenfold red, orange, yellow, green, blue, indigo and violet,
A week has 7 days. Our world has 7 wonders.
> We start from the reciprocal of 7.
1 / 7 = 142857 / 999999 = 0.142857142857141857 .. .. .. (repeating forever)
The string 142857 is fairly interesting as shown below:
1 × 142857 = 142857
2 × 142857 = 285714
3 × 142857 = 428571
4 × 142857 = 571428
5 × 142857 = 714285
6 × 142857 = 857142
Now see the amazing pattern at the side: it always follows the same pattern 1 -> 4 -> 2 -> 8 -> 5 -> 7; (when you’ve come to the “7”, come to “1”), the only difference is starting with different first digit. It’s a full-circle pattern, as shown.
It’s also interesting to note that the digits 3, 6, 9 are skipped in each of the strings. One explanation is for the first digits 14, 2 × 14 = 28 and 3 × 14 = 42 so there is no thirty something for the first two digits of any string. Of course 3 × 13 = 39 but no string starting with 13 will have nice pattern like that 142857.
> Suppose you are rolling standard 6-sided dice with numbers 1- 6 on respective faces. If you roll two dices independently, and then add the number of the top face of each dice to get the sum. What would be most likely for this sum to be? You guessed it – “7” (as appeared in combination 1 & 6, 2 & 5, 3 & 4, 4 & 3, 5 & 2, 6 & 1) – a total of (1/6) likelihood.
How many different numbers can we get? From 2 to 12: there are 11 numbers. If you do the experiment 110 times, in about 18 times you’ll get the sum “7”, but on average since there are 11 numbers, each number has about 10 chances. Now you see the significant difference.
In actual experiment, the results may vary as we always in experiment with random events.
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(the Kissing Solution for a Unit Circle).
The problem is to use several circles of diameter 1 (the smaller circles) to completely cover a perfect circle of diameter 2 (the large circle, which has twice as long radius, and thus is four times in area).
Now at least how many smaller circles do we need to completely cover the large circle? And how?The answer is that you need at least 7 circles.
Can you figure out how? And Can you show less (say 6) of smaller circles are not enough to cover the large circle?
> 7 is a number in the Mersenne sequence, which goes as:
21 – 1, 22 – 1, 23 – 1, 24 – 1, 25 – 1, 26 – 1, 27 – 1, .. .. ..
Or
1, 3, 7, 15, 31, 63, 127, .. .. ..
Of the Mersenne sequence, number 7 is the third number . Both 3 (from the 3rd) and 7 are primes.
It has been proven that if n is NOT a prime number, then 2n – 1 is not a prime either. How about the other side? If n is a prime number, does that guarantee that 2n – 1 is also a prime. This is a famous conjecture about the Mersenne primes; but today we know that it is not so. We refrain from discussing here, but that’s an interesting topic.
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If we take the quadruple of 7, we get 28. Since
28 = 1 + 2 + 4 + 7 + 14
(Notice that the right hand has included, as the addends, all the proper divisors of 28)
So we can conclude that 28 is a complete number. It is 2nd complete number, coming after 6.
> Number 7 is also be considered as “mystic” or “magic” in both oriental and western cultures. In English, “7” sounds like “heaven”. Number “7” has been used many times in the bible, for example, the great flood lasted 7 weeks (7 x 7 days), and Egypt had 7 abundant years followed by 7 disastrous when all harvest and so bad, etc.
We have mentioned 7 wonders of the world . And have you heard the English phrase “at sixes and sevens”? It is a British phrase used to describe a state of confusion or disarray – – so not good — But whether good or bad, “7” is special.
