Open Writing

I play with a new type of Arithmetic that I call THRITHMETIC. It is (I think) quite useless but can give you a lot of fun if you are mathematically inclined. It is based on the number 3. It has its own vocabulary which I will explain.

In normal Arithmetic, numbers are either EVEN, which means that they can be exactly divided by two without any remainder, or they are ODD, which means that they are not divisible by two without remainder. Another way of expressing this is to say that ODD numbers are divisible by two with a remainder of ONE. All numbers are either EVEN or ODD. No other numbers exist.

Similarly in THRITHMETIC, numbers are either THREEVEN (exactly divisible by three without a remainder) or they are not THREEVEN. There are two reasons why a number can be not threeven. Firstly, when divided by three, it may have a remainder of 1 in which case we call it THRODD or alternatively on division by three there may be a remainder of two in which case we call it THROODD. All Numbers must either be THREEVEN, THRODD or THROOD.

In the simples terms then
3, 6, 9 and 12 are THREEVEN
4, 7 10 and 13 are THRODD
5, 8, 11 and 14 are THROOD
These definitions may be continued to all magnitude of numbers and for example:-
100 is throdd since 100 / 3 = 33 Rem 1
254 is throodd since 254 / 3 = 84 Rem 2
573 is threeven since 573 / 3 = 191 No Rem.

The easy way to determine whether larger numbers are threeven, throdd or throodd is to add up their individual component digits until a single digit answer is arrived at. It is then an easy matter to decide whether the single digit is threeven, throdd or throodd, and this in turn means that the original number is similarly, threeven, throdd or throodd.
Examples are :-
108 1+0+8= 9, 9 is threeven ˆ108 is threeven
156 1+5+6= 12, & 1+2= 3, 3 is threeven ˆ 156 is threeven
276 2+7+6= 15 & 1+5= 6 6 is threeven ˆ 276 is threeven
247 2+4+7= 13 & 1+3= 4 4 is throdd ˆ 247 is throdd
616 9+9+7= 15 & 2+5= 7 7 is throdd ˆ 616 id throdd
437 4+3+7= 14 & 1+4= 5 5 is throodd ˆ 437 is throodd
980 9+8+0= 17 &1+7= 8 8 is throodd ˆ 990 is throodd

To make matters even easier, you may omit any of the digits 3, 6 or 9 whilst adding since they make no difference to final result.

Looking into this system, some interesting patterns occur if you add numbers together:-
1 If you add two threeven numbers together, the result is always threeven.
2 If you add a threeven number to a throdd number the result is always throdd.
3 If you add a threeven number to a throodd number the result is always throodd.

Summarizing, 1, 2 & 3 above if you add a threeven number to any other number the resulting number is in the same category as the original number.

4 If you add two throdd numbers together the result is always throodd: -
5 If you add two throodd numbers together the result is always throdd
6 If you add a throodd to a throdd you get a threeven.
Examples are:-
Addtogether 9threeven 13throdd 23throodd
12threeven 21threeven 25throdd 35throodd
19Throdd 28throdd 32throodd 42threeven
8throodd 17throodd 21threeven 31throdd

Interesting patterns also occur if you MULTPLY numbers together:-
Examples are:-
MultiplyTogether 9threeven 13throdd 23throodd
12threeven 108threeven 156threeven 276threeven
19throdd 361threeven 247throdd 437throodd
8throodd 72threeven 104throodd 184throdd

These are patterns are even easier to summarise.
1) Multiply any number by a threeven number and the result is threeven.
2) Multiply any throdd by a throdd or a throodd by a throodd and the result is throd.
3) Multiply a throdd by a throodd and the answer is throod.
Other interesting patterns also arise:-
1) Take any three numbers consisting of three consecutive digits, add them together and the result is always threeven.
Eg
234 = 2+3+4 = 9 = Threeven
678 = 6+7+8 = 21, and 2+1 = 3 = Threeven
99100101 =99+100+101= 300, and 3+0+0= 3 = Threeven.

Of course this also applies, regardless of the order of the digits in the starting number.
So 234, 243, 324, 342, 423 and 432 are all Threeven.
And 678, 687,768, 786, 867 & 786 are all threeven
And remembering that you can omit any of the threeven number in the count up
24 and 42 are also threeven as are 78 & 87
and taking this through to the 99,100, and 101 example
100011, 101010, 111000 are all threeven numbers as of course is 111.

If you think it’s fun playing with thrithmetic, then even more fun can be had if you consider looking at the number three cubed which is of course 9, and a very powerful player.
1) Take ANY number whatsoever e.g my telephone number 711619, then reverse it and take the larger from the smaller
916117 - 711617 = 204498
Add together the digits of the answer 2+4+4+9+8 = 27
2+7 = 9
It always 9, regardless of the starting number except when the digits are the same eg 444 – 444 = 0
Example 2
987654321 – 123456789 = 864197532
8+6+4+1+9+7+5+3+2 = 45 and 4+5 = 9
Example 3
8463921 - 1293648 = 735273
7+3+5+2+7+3 = 27 and 2+7 = 9

I just wonder what happens if we were to start looking at Pentithritic