Challenge Level

A group from St. Lukes School in Cambridge had a go. They were Saul, Tom, Kain, Emma, Stephanie, Sara, Sophie, Jennifer and Mariko, and here are some of their results, also showing the Digital Roots alongside.

Saul did this using 6 - 3 x 8;

and he noticed that the units went 6,4,8,0, and the DigitalRoots were 6

Emma did 2, firstly 9 x 3 + 1, then 10 x 2 - 3;

and noticed the repeats in the units columns.

Jennifer did this by using 6 x 4 - 4;

she noticed th 6,0, repeating in the units

Sophie did this using 4 x 3 + 8;

she noticed the DigitalRoots settled into all 5's, and there's a units pattern.

Mariko used 9 x 3 + 9;

She noticed that the units goes 9,6,7,0, also that all the numbers were divisible by the starting number, she noticed that all the numbers are divisible by 3, she thought that this is because 3 is divisble into 9 and 9 is a factor of all the numbers.

Tom explored 4 x 9 - 13

Sara looked at 3 x 5 - 2

she noted the units being 3's and the tens going 1,6, the hundreds going 3,5,8,0. I know that this is unusual so I looked at the thousands going further as 1,7,9,5,6,2,4,0 and repeats.

Stephanie used 4 x 7 - 10

she noticed that the units doubled from 4 to 8 to 6 to 2 then go back to 4

Thank you St. Lukes for this really good work, well done and keep on with your work sending in your solutions to the*live* problems.

One pupil from another school (don't know which) explored a great deal and here we summarize what was found :-

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

Saul did this using 6 - 3 x 8;

and he noticed that the units went 6,4,8,0, and the DigitalRoots were 6

Emma did 2, firstly 9 x 3 + 1, then 10 x 2 - 3;

and noticed the repeats in the units columns.

Jennifer did this by using 6 x 4 - 4;

she noticed th 6,0, repeating in the units

Sophie did this using 4 x 3 + 8;

she noticed the DigitalRoots settled into all 5's, and there's a units pattern.

Mariko used 9 x 3 + 9;

She noticed that the units goes 9,6,7,0, also that all the numbers were divisible by the starting number, she noticed that all the numbers are divisible by 3, she thought that this is because 3 is divisble into 9 and 9 is a factor of all the numbers.

Tom explored 4 x 9 - 13

Sara looked at 3 x 5 - 2

she noted the units being 3's and the tens going 1,6, the hundreds going 3,5,8,0. I know that this is unusual so I looked at the thousands going further as 1,7,9,5,6,2,4,0 and repeats.

Stephanie used 4 x 7 - 10

she noticed that the units doubled from 4 to 8 to 6 to 2 then go back to 4

Thank you St. Lukes for this really good work, well done and keep on with your work sending in your solutions to the

One pupil from another school (don't know which) explored a great deal and here we summarize what was found :-

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