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I woke up. It was a cloudy afternoon. I lived a grey life. But it hadn't always been that way.

There was a girl, back home. Lin. She could steal the sun out of the sky. I almost proposed. But I've heard it said that when fate comes knocking, it's too late to board your windows. Cue the draft, and I thought I'd never see her again.

There was a younger guy in my squadron. Mikey. He was alright. He thought he was a mathematician. I reckon he was right. Most days he'd bust in and tell me... I don't know. How unlucky I was, being born when I was born. How I was statistically more likely to be drafted. Like it mattered then.

Some days he'd have interesting war math, he got his info from some mole on the other side. Turns out the opposing army received one new squadron of 50 soldiers every five weeks, and each soldier received one automatic rifle and 100 rounds on the first day, and for every "tomorrow", they received 90% of what they received "today". For example, on day one every soldier out there got 100 rounds. The next day? 90. The day after that? 81. after that? 72. point 9. you can't fire point nine of a bullet, so we'll call it 72... as a general rule it works. Oh yea, and every 3 weeks there would be a new truck full of ammunition earmarked for a particular squadron so that squadron's cycle started over.

So here's a question for you. I'd been out here for five months. How many bullets came at me and my men? If you have a calculator I'll walk you through it. Set that thing to sequences and head on over to your "y equals" screen. bear with me. If they got 90% of their previous days allotment, they got "nine tenths multiplied by u of n minus one". The brackets keep your fractions and terms separate from each other. set your "u of n-min" to 100 and head on back to the home screen. u(1) is your first day. That'll be 100, you told it to say that. u(2), 90. We're in business. If they get a new truck in three week's time, that's u(1,21) or the period during which the pattern is in effect. Store that in list 1.

Go to list, math, sum. enter list 1. Store that in Alpha A. 890.58. 890. Thinks that's bad enough? That's only three. weeks. Keep in mind, if one of them went down, the rounds don't disappear. somebody else would just pick them up.

How many squadrons came in the five months I spent here? that's 20 weeks. 20 divided by 5, the number of weeks it takes for the next squadron to arrive, is four. So you'd have to find the number of ammo cycles that each of the four squadrons received, and when each of them arrived. lucky you, I already found that for ya. Here it is. First squadron, 20 weeks. Second squadron, 15 weeks. Third squadron, 10 weeks. Fourth squadron, 5 weeks. There's an ammo cycle every 3 weeks, and each cycle comprises 890.58 rounds.

For the next step, we're going to take each squadron's on-duty time, divide it by the time it takes for a shipment to arrive, and multiply that by the amount of ammo in a shipment. We saved that in A. Take each answer and store them in B,C,D, and E respectively.

1. 20 ÷ 3 X A = 5937.206... stored in B.

2. 15 ÷ 3 X A = 4452.905... stored in C.

3. 10 ÷ 3 X A = 2968.603... stored in D.

4. 5 ÷ 3 X A = 1484.301... stored in E.

So, curious what B+C+D+E was? (fourteen thousand eight hundred and 43 point oh one six.)14843.016... But there were 50 combatants per battalion, so times 50 is (seven hundred and forty two thousand, one hundred and fifty point eight four.) 742150.84, drop the point 84. 742150. Mikey was wrong. Unlucky about my birth? No way, I'm lucky just to be alive. But a few days back one of those got me pretty bad. They said I might not walk again. I told them I just wanted to go home.

That all seems like an eternity ago, and I couldn't be happier. You see, there's this girl. Lin. And I think it's about time I paid her a visit. I'm home, and I won't make her wait another minute.

So let's take a look at a sketch I drew up of this place and figure out the fastest way there. Think of the nodes as intersections or destinations. The lines are roads. Here's the airport, that's where I am. She lives a town over. I definitely need to catch a train if I want to make it over there quick. There's the train station and all the ways there. There's only one way to take a train, so I won't both showing the gap between towns. Here's the station in her town, and her place is just down that way.

Like a train, there's only one way to travel in a straight line. But every turn you take, there's plenty of ways to get there. Two ways to get here, and six ways to get here. Here's the kicker. To go three down and three across, there's twenty options. Quick tip. If you go one across, you just count up. 1,2,3,4,5. To find the sum inside any node, just add the values of the nodes that lead up to it.

I've got to make a few stop along the way. Flowers here, Chocolate here. And in the next town over, I'm picking up the ring. Tonight's the night, So let's get started. First stop is easy, I already showed you the pattern. 5 ways. The next two, if you look carefully, are both 3X3 grids. 20 options. But here's a quick way to figure that out. 123, plus 123, 6. 4 plus 6, 10. 10 and 10 is twenty. Let's get on that train. The Jeweler is two streets away and one down, so count it, 123. one more down and four across, so four it is. Only one last thing to find.

How many ways could I get there? Let's code our stops. F for flourist, C for chocolate shop. T for train, J for Jeweler and L for Lin's. Remember to multiply, because whether you take the blue or the black path is a different journey from A to B. The two A paths by the three B paths is how you find all the AB paths.

So what is FCTJL or (5)(20)(20)(3)(4) is 24000. Next time you think you're out low on options, remember that mixing and matching makes a world of difference.

Well, my cab is here. Time to brighten this grey world of mine.

Answers

Squads of 50 soldiers receive ammunition at a rate of (9/10)*u(n-1) [nMin:100] or nine tenths the previous day's allotment per day with 100 on the first day. With shipments every three weeks which restart the cycle, as well as another new 50 soldiers joining the fray every five weeks, how many bullets are fired from that army in 5 months?

A= u(1,21)

1. 20 ÷ 3 X A = 5937.206... aka B.

2. 15 ÷ 3 X A = 4452.905... aka C.

3. 10 ÷ 3 X A = 2968.603... aka D.

4. 5 ÷ 3 X A = 1484.301... aka E

50(B+C+D+E) = 742150.84

How many ways can one travel a path which is constructed of five grids, if said five grids are of dimensions 1X5, 3X3, 3X3, 1X2, and 1X3 respectively? Using Pascal's Triangle Arithmetique, one can easily deduce that said grids are comprised of 5 paths, 20 paths, 20 paths, 3 paths, and 4 paths respectively. Because a journey stretches from beginning to end, and a different path on any of the five grids results in a different total journey, the values of the grids must be multiplied by each other. (5)(20)(20)(3)(4) is equal to 24000.

Reflection

I chose the concepts I chose because I thought it would be interesting to illustrate a man detached from reality but obsessed with facts. It's hard to explain... He meticulously calculates the number of bullets which were probably fired at his army without really seeming to understand the cause and effect of such a thing. The next problem is also interesting, he doesn't seem particularly concerned with how he gets to his destination as long as he makes a few select stops, yet he still calculates all of the possibilities. A fictional character performing a thought experiment.

I constructed questions I believed were slightly above the level we covered in class. Instead of a pattern, I constructed a layered pattern, almost akin to a fractal. While before we only went so far as to calculate two grid paths, I chose to make a five-grid path. The concept is the same, but it's interesting how quickly the additional grids add up.

Because I was teaching myself, it's not accurate to say that I learned anything. I did however hone previously acquired skills and reinforce previous knowledge. For example, I spent ages trying to figure out why my equation for the ammunition depletion wasn't working before it dawned on me that I was missing the "u" before (n-1). That in itself helped me to understand the value of this exercise.

In the future, I would have given the students one class in which to work on this assignment. One class wouldn't be enough to get it done, but if the class were structured properly it could spur some of the more lethargic students to continue on their own time. I for example was putting this off until I was comfortable with the assignment, then I realized that I would have to make it comfortable because KUROPATWA DOESN'T SPOONFEED. I would actually like to thank you for that Mr. K, because I appreciated my own effort.