Taller Than Average Tales: Four for four

10 March 2008

Four for four

The workout challenge/experiment is going well so far. You may recall that I said if I miss a day for any reason, I'll add 3 more consecutive days to the challenge, and that if I blow off more than one day in a row, the number of make-up days will increase by 1 for each consecutive day missed.

Then, I gave the following examples:
If I'm sick three days in a row, I'll add 9 days to the challenge (3+3+3).
If I blow off three days in a row, I'll add 12 days to the challenge (3+4+5).

Since I'm still a long way from 30, I started wondering how many make-up days I was saving myself by sticking to the plan. In the worst case scenario, I would've missed all 30 days by never actually starting.

y=x+2 where x is the last consecutive day missed and y is the number of makeup days for that day. So by working out the first four days in a row, I've saved myself from missing 30, 29, 28, and 27 consecutive days. That means I've saved myself from:

(30+2)+(29+2)+(28+2)+(27+2)= 32+31+30+29 = 122 makeup days, or 4 extra months.

That's worth being stoked about.

Relying on examples to explain how makeup days would be calculated, however, felt lame, so I started working out a formula for it. My break through came hours before I intended to get up this morning, and it inspired me enough that I couldn't get back to sleep until I'd figured it out.

I recalled the story about the student who was disrupting his math class. To keep him occupied and quiet, his teacher told him to add up all the numbers between 1 and 100. He finished in about five minutes because he recognized that 1+100=101, 2+99=101, 3+98=101, and so on, up to 50+51=101. Since there were 50 pairs of numbers, all he had to do was multiply 50*101 to get the answer, 5050.

My next step was to develop a formula that would provide the sum
of any series of Numbers starting with one. I came up with two equations, one for a series ending in an even number and another for those ending with an odd number:

The total sum of a series of numbers where the final number, n, is even:

The total sum of a series of numbers where the final number, n, is odd:

Then it was just a matter of adapting those formulae for a sequence that begins with 3 instead of 1. In the following equations, n is the number of consecutive workout days blown off.

When n is even,