The Magnificent 1/7th

The Magnificent 7. Seven Sisters. Seven Police Academy movies. Seven deadly sins.  Seven  Dwarfs. Seven Brides For Seven Brothers.

I was watching a show last night, when a character complained about his 10 days of annual leave a year said, 'It's only 1.426 weeks of year.' I called out 'It's 1.429!' Or to be more precise 1.428571.

So if you ever wanted to know how much of the the deadly sins gluttony makes up or what percentage Dopey, Doc & Sneezy are of the Seven Dwarfs you're in luck.

1/7 expressed as a fraction is .142857142857...

It keeps on repeating in the same pattern.

The first thing that struck me is that the 6 different digits in the sequence does not include a zero or any multiple of 3.

Also if you add the 1st & 4th digits, the 2nd & the 5th or the 3rd & the 6th digits, the answer is always 9. It turns out there is a good reason for this.

The full sequence of the fractions is:

1/7 - .142857
2/7 - .285714
3/7 - .428571
4/7 - .571428
5/7 - .714285
6/7 - .857142

So the 6 digits are used in all of the fractions. Also the order that they appear in does not change. The digit that starts the fraction changes, but the sequence stays the same.

It also means every digit is in a different slot for each fraction. Eg 1 is 1st in 1/7, 5th for 2/7, 3rd for 3/7, 3rd for 4/7, 2nd for 5/7, and 4th for 6/7.

Also if you look at the first 3 digits & the last 3 digits you'll notice that for;

- 1/7 & 6/7
- 2/7 & 5/7
- 3/7 & 4/7

just switch around the block of three digits and you'll get the answer to the other. Eg 1/7 is .[142][857] & 6/7 is .[857][142].

Also if you double the first 2 digits of any sequence you will get the answer is to what the next 2 digits are, (in some cases you have to add 1 to the answer if it breaks certain rules).

So for 1/7 - 14 doubled is 28. Them 28 doubled is 56. Now there are no multiples of 3 in the sequence, so just move up one number to 57. Put that together and we get 142857.
The double of 57 is 114. Take the last 2 digits of that which is 14 and you're back to the start of the sequence.

For 5/7 - if you double 71 you get 142. The last 2 digits are 42. If you double 42 you get 84. Now we've already used a 4 so we move up a number to get 85. So 714285 Double 85 & you get 170. The last 2 digits are 70. Now as we know we can't have a zero, so we move along 1 to 71. And we are back to the start of the sequence.

The 1/7 fraction. The most beautiful number there is, just ahead of 867-5309.