Primer-The W Sequence

Even and Odd Numbers

Even numbers are the numbers of sharing jelly beans among two friends. You make sure that they are split evenly, grabbing two at a time and dropping one in each bowl. This is what it means for a number to be even. If you placed both bowls on the opposite ends of an old-fashioned scale, then they would balance. That is unless you accidentally drop an extra jelly bean in your own bowl, tipping the scales so to speak in your favor; then you have an odd number of jelly beans. This is what it means for a number to be odd.

The first five even numbers are 0, 2, 4, 6, and 8. The first five odd numbers are 1, 3, 5, 7, and 9. As martian mathematics is largely single digit in nature, these are the only ones that matter here. It is a happy coincidence that the evenness or oddness of a number—called parity—is betrayed by the ones' place value of the number in question (I will show why this is in a later post): 332 is even, while 241 is odd. 

Knowing the even and odd digits is enough to add or subtract two. The sum or difference will just be the next or previous even or odd number, with adjustments in the tens' place as appropriate. I will leave it to you to put this to practice. The diagram below should help, however.

We can also add or subtract eight using the even and odd sequences because of the complementary relationship: 8 = 10 – 2. If we are adding eight then we move through the sequence backwards and always increase the tens place, except when we cycle back around. For example, 37 + 8 = 45. We do the opposite when we subtract, moving forwards instead of backwards and decrementing the tens' place instead of incrementing. For example, 29 – 8 = 21. We did something similar when adding and subtracting seven. 

Organizing the even and odd sequences into a 'W' pattern makes it possible to easily add and subtract four. I call this the 'W' sequence, because of the 'W.' Let's skip count in fours starting at zero to see how the 'W' sequence works:

0, 4, 8, 12, 16, 20

We increment the tens' place when changing rows. When counting backwards, we would obviously decrement the tens' place instead. The odd 'W' sequence works exactly the same. Now we can easily add or subtract four from any number by navigating the sequence. For example, 78 + 4 = 82 since 2 comes after 8 in the sequence on the next row. Similarly 83 – 4 = 79. Since the 'W' sequence has a clearly defined beginning, middle, and end, it is easy to visualize in your head. With practice, adding and subtracting 4 from any number should be effortless.

As usual, we can extract more utility from the 'W' sequence through its complementary relationship: 6 = 10 – 4. Let's skip count in s
ix starting at zero to see how the 'W' sequence works:

0, 6, 12, 18, 24, 30

As with all the complementary relations, we change direction for addition and subtraction and always increment or decrement the tens' place except when we change rows. For example, 58 + 6 = 64 since adding moves backwards to the previous number in the sequence, 4, and the tens' place value increases because we are adding and because 4 and 8 are on the same row. Similarly, 37 – 6 = 31 since subtracting moves forwards to the next number in the sequence, 1, and the tens' place value is unchanged because of the row change. 

You will find that on some problems reverting to your old way of doing it is faster and easier. That's fine. A helpful hint regardless of how you do it is to think of –6 problems as +4 problems and +6 problems as –4 problems. Of course, the tens' place values are different, but sometimes adding four to the ones place is easier than subtracting 6. For example, 32 – 6 = 26 (2 + 4 = 6) or 35 + 6 = 41 (5 – 4 = 1). This works the other way around too. If you have a problem where subtracting four would be harder than adding six then just adjust the tens' place using common sense and add six to the ones' place instead. For example, 42 – 4 = 38 (2 + 6 = 8) or 49 + 4 = 53 (9 – 6 = 3). 

Here are some practice problems for your enjoyment.

This concludes all the background information on addition and subtraction needed for martian mathematics. In my next post, we will begin with the core of martian mathematics—doubling.
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