Exploring Wild & Wonderful Number Patterns

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Number Pattern Explorations

Stage: 2 and 3 Challenge Level:

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 :-

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.

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!