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'Exploring Wild & Wonderful Number Patterns' printed from http://nrich.maths.org/

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Here is a little journey taken by one creative explorer:-

They decided on an easy start by taking:



3 x 8 + 4

The pattern went as follows:-



2 8
2 2 8
1 8 2 8
1 4 6 2 8
1 1 7 0 2 8
9 3 6 2 2 8
7 4 8 9 8 2 8
5 9 9 1 8 6 2 8
4 7 9 3 4 9 0 2 8

This was quite nice, after the first number [the chosen one] all the digits in the units column are 8, the tens are 2 and the digital roots go:- 1, 3, 1, 3, 1, 3, 1, 3, 1.

So what happened when the starting number is changed?



4 x 8 + 4


 
4
3 6
2 9 2
2 3 4 0
1 8 7 2 4
1 4 9 7 9 6
1 1 9 8 3 7 2
9 5 8 6 9 8 0

And the digital roots now go:- 9, 4, 9, 4, 9, 4, 9, 4

Whoa! That's quite something. The units now go, after the first chosen number, 6204 and repeat.

Well let's summarize what the rest of the exploration gave :-

RULE UNITS DIG.ROOTS OTHER COMMENTS
3 x 8 + 4 8.... 1 3 1 3 1 3 ... Tens > 2 Hunds > 2860....
4 x 8 + 4 6204 ... 9 4 9 4 9 4 ...
5 x 8 + 4 4260 ... 8 5 8 5 8 5 ...
6 x 8 + 4 2046 ... 7 6 7 6 7 6 ...
7 x 8 + 4 0462 ... 6 7 6 7 6 7 ...
8 x 8 + 4 8... 5 8 5 8 5 8 ... Tens > 6480
9 x 8 + 4 6204 ... 4 9 4 9 4 9 ...
10 x 8 + 4 4620 ... 3 1 3 1 3 1 ... So we could predict that the next ones go:-
11 x 8 + 4 2 ... 2 ... Dig. Roots should be 2, 11 but 11 is 2 so all are 2!
12 x 8 + 4 0462 ... 1 3 1 3 1 3 ...
13 x 8 + 4 8 ... 9 4 9 4 9 4 ... Checked, and Tens > 0648....

Well reading this has been very exciting and led Bernard to make a further change, using 7 instead of 8.

RULE UNITS DIG.ROOTS OTHER COMMENTS
2 x 7 + 4 8024... 945378612 .. Tens > 1310 ...
3 x 7 + 4 5973... 786129453 .. Tens > 2750 ...
4 x 7 + 4 2804... 537861294 .. Tens > 3200 ... Hunds > All even
5 x 7 + 4 9735... 378612945 .. Tens > 3740 ...
6 x 7 + 4 6... 129453786 .. Tens > 4280 ... Hunds > 3200...
7 x 7 + 4 3597... 861294537 .. Tens > 5720 ...
8 x 7 + 4 0428... 612945378 .. Tens > 6270 ...
9 x 7 + 4 7359... 453786129 .. Tens > 6710 ...
10 x 7 + 4 4280... 294537861 .. Tens > 7251 ...
11 x 7 + 4 1... 945378612 .. Tens > 8701 ... Hunds > 5000 Thous > Even

There is so much here that is fascinating.

I hope that your explorations have gone well!