Completing and beginning, summing up, and miscellany.
Inside: exercises, “math”, and the day in {re|pre}view.
I.
At the beginning of the year I did the ‘January Jumpstart’, and did a mix of cardio and various videos over the course of four weeks. It was at times difficult to stick with it, depending on my energy level, but I was pleased when and that I did.
When February rolled around I knew I needed a different, short-term, measurable challenge, and so I turned to both ‘one hundred push ups‘ and ‘two hundred squats’; in one sense it’s just a poorly thought out ‘program’ designed to help the site’s owner make money off the ‘book’ (such as it is) and the app, though I cannot begrudge someone that.
Yesterday I completed both the push up and squat program, the former in the morning, the latter at night. Afterward my legs, especially, felt wobbly and rubbery. I was not that surprised that I could complete the squat test on schedule, but push ups and other upper body (but not necessarily arm) exertions have always been my weakness, especially as I’ve always had long legs and have carried much of my weight in my legs. I still cannot do pullups, and haven’t been able to do any since elementary school.
I began the push up program in the right-most column, but just barely, and I just barely managed to remain there after every ‘test’ along the way. Yet I’m astonished at how quickly and steadily I did progress, especially since I began with being able to do only about fifteen or so push ups in a row, and a month and a half later made it to one hundred, though not with much room for error.
Now I need to work on form and precision, on being able to do different kinds of push ups reliably and well, and so on.
And then I thought to myself, what next?
It’s been years since the dreaded and infamous squat-thrusts of high school P.E. and track practice, but my thoughts do turn to the burpee. And just as my thoughts turned to the burpee, and just as when my thoughts turned to roasting cowpeas and lentils I discovered that roasting lentils was our ‘new thing’ in 2013 (after roasting chickpeas in 2012, after banana soft serve was our ‘thing’ a couple years ago, and so on), I discover that ‘100 burpees’ is the new ‘thing’ for early 2013.
See also:
- The Burpee Workout
- http://WOW: Cave Burpees
- Fitness – Boot Camp Workout 1: Burpee’s Exercise
- 100 Day Burpee Challenge
- Day 100
In the last link, our video informs us that 5050 burpees were performed.
Yes they were.
Aside: while burpees are currently our 2013 ‘thing’, the ‘100 Day Burpee Challenge’ goes back at least to 2011, as evidenced by a 2011 HuffPo article.
II.
As soon as I saw 5050, my immediate response, partially intuitive and almost instinctual, was ‘that is correct’.
And I was reminded of a simple and elegant basic result from mathematics, illustrated here as the sum of 1 through 100 is 5050. That is, one plus two plus three plus … up to one hundred.
So what, in general, is the sum of 1 through ‘n’, where ‘n’ is an integer greater than 1? We begin with a few examples. 1 through 1, is just 1. 1 to 2; that’s 1 + 2 = 3.
1 = 1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
…
But continuing doing this won’t necessarily get us an easy, compact answer, a simple formula into which we plug ‘n’ and get a result. But let us make an observation. I want to sum 1 through 10. I notice that 1 + 10 = 11, and also that 2 + 9 = 11, and 3 + 8 = 8, 4 + 7 = 11, and 5 + 6 = 11. I can always match the lowest and the highest, the next lowest and next highest, and so on. If I have an even number (such as summing 1 to 10, 1 to 20, 1 to 36, etc.), I get do divide my numbers in half and pair them up, but if I’m summing 1 to an odd number, there’s a number in the middle left over. How many pairs do I get? If I go 1 to 10, I get 5, if I go 1 to 20, I get 10, and so on, that is, n/2. So I have n/2 pairs, and if I add them together (each of which is equal to n+1), I get the sum of 1 to n. That is, I suspect that 1 + 2 + … + n = (n+1)*(n/2), or (n(n+1))/2, or (n^2+n)/2 … however you want to write it.
This isn’t a proof, but one can test a few cases to see that they work, and then one can use mathematical induction to provide an actual proof (you have to show that if it works for n, it then works for n+1).
The proof is formal and logical; the manner of solving it presented above is rather intuitive. The story goes that Gauss’ teacher presented the ‘add the integers from one to one hundred’ problem to his class, expecting them to take a long time finding answers, and Gauss presented a solution after only a few minutes; he was but a child.
We also call these numbers — 1, 3, 6, 10, 15, 21, 28, … — ‘triangular numbers‘, and they give us the binomial coefficients, which for n give us the number of distinct pairs that can be selected from n+1 objects. Here we say ‘n plus one choose two’.
III
Today I opened up a can of herring in paprika sauce by Polar; it was a nice, long, rectangular tin with rounded corners. I ate a couple from the can, other bits and pieces I put on a slice of dense rye bread I had available. It all went well with grape tomatoes and sliced dill pickles.
Tonight we get “The Curse of Frank Black” and “Christmas Carol” in ‘Millennium’ and ‘X-Files’, respectively.
On Tuesdays we often go out for beer downtown and before that to a movie at the art-house series, but this week’s entry, a documentary, doesn’t appeal to us as a watch-in-a-group event.
There has been no stomping around upstairs today … it’s as if she reads this blog and chose to modify her — or that child’s — behavior! Or else the child just isn’t there anymore. One can hope.
