Tag Archives: Open

A new toy problem

First, a short math question from Peter:

Question: What is the coefficient of $latex x^{12}$ in the simplified expression of $latex (a-x)(b-x) \dots (z-x)$?

I often hate these questions, but this one gave me a laugh. Perhaps it was just at the right time.

A police car passed me the other day with sirens wailing and I became reminded full on about the Doppler Effect. But the siren happened to agree with a song I was whistling to myself at the time, and this made me wonder – suppose we had a piece of music (or to start, a scale) that we wanted to hear, and we stood in the middle of a perfectly straight train track. Now suppose the train had on it a very loud (so that we could hear it no matter how far away it was) siren that always held the same pitch. If the train moved so that via the Doppler effect, we heard the song (or scale), what would its possible movements look like? How far away would it have to be to not run us over?

Some annoying things come up like the continuity of the velocity and pitch, so we might further specify that we have some sort of time interval. So we have a scale, and the note changes every second. And perhaps we want the train to have the exact right pitch at the start of every second (so that it would have constant acceleration, I believe – not so exciting). Or perhaps we are a bit looser, and demand only that the train hit the correct pitch each second. Or perhaps we let it have instantaneous acceleration – I haven’t played with the problem yet, so I don’t really know. I’m just throwing out the idea because I liked it, and perhaps I’ll play with it sometime soon.

Now, the reason I like it is because we can go up a level. Suppose we have a car instead, and we’re in an infinitely large, empty, parking lot (or perhaps not empty – that’d be interesting). Suppose the car had a siren that wailed a constant pitch, too. What do the possible paths of the car look like? How does one minimize the distance the car travels, or how far from us it gets, or how fast it must go (ok – this one isn’t as hard as the previous 2)? It’s more interesting, because there’s this whole other dimension thing going on.

And even better: what about a plane? I sit on the ground and a plane flies overhead. What do its paths look like?

All together, this sounds like there could be a reasonable approach to some aspect of this problem. Under the name, “Planes, trains, and automobiles” – or perhaps in order – “Trains, automobiles, and planes,” this could be a humorous and fun article for something like Mathematics Magazine or AMMonthly. Or it might be really hard. I don’t know – I haven’t played with it yet. I can only play with so many things at a time, after all.

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A Question on Stoplights

Consider stopping at a traffic light: the normal way of things is that all the cars are packed tightly together. Decorum decrees that one waits for the car ahead to accelerate at a green light, lest accidents occur. But is that really necessary? What if instead, the cars left somewhere between a half and a full car length between them so that they could all accelerate at a green light (and still have enough time to slow down/stop if they accelerate too quickly)? Is this efficient or not? Is there any change in traffic flow?

My thinking is that in long lines, to have every car accelerate at the (almost) same time would yield faster traffic flow, but perhaps that’s misguided. But I’m thinking of doing a simple study of the idea, just to see what happens.

Happy weekend!

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Containers of Water: Maybe an interesting question.

Consider the old middle-school type puzzle question: Can you measure 6 quarts of water using only a 4 quart bottle and a 9 quart bottle? Yes, you can, if you’re witty. Fill the 9 quart bottle. Then fill the 4 quart bottle from the 9 quart bottle, leaving 5 quarts in the 9-bottle. Empty the 4-bottle, and fill it again from the 9-bottle, leaving only 1 quart in the 9-bottle. Again empty the 4-bottle, and pour the 1 quart from the 9-bottle into it. Now fill the 9-bottle one last time and pour into the 4-bottle until it’s full – at this time, the 4-bottle has 4 quarts and the 9 bottle has 6 quarts. Aha!

But consider the slightly broader question: how many values can you get? We see we can get 1, 4, 5, 6, and 9 already. We can clearly get 8 by filling the 4-bottle twice and pouring this bottle into the 9. With this, we could get 4 by filling the 4-bottle again and then pouring as much as possible into the 9-bottle when its filled with 8 – as this leaves 3 quarts in the 4-bottle. Finally, we can get 2 by taking 6 quarts and trying to pour it into an empty 4-bottle. (With 2, we get 7 by putting the 2 into the 4-bottle and then trying to pour a full 9-bottle into the 4-bottle). So we can get all numbers from 1 to 9. If we were really picky, we could note that we can then easily extend this to getting all numbers from 0 to 13 quarts, albeit not all in one container.

If you go through these, it turns out it is possible to get any number between 0 and 13 quarts with a 4-bottle and a 9-bottle in at most 10 pours, where a pour includes filling a bottle from the water source (presumably infinite), pouring from one bottle into another, or emptying the water into the water source. Now I propose the following question:

Given only 2 bottles (of an integer size) and up to 10 pours, what is the largest N s.t. we can measure out 0, … , N quarts (inclusive)?

As is natural, let’s extend this a bit further. For any subset of the natural numbers $latex S$ and for a number of pours $latex p$, define

$latex {\bf F}(S; p) :=$ the set $latex R$ of possible results after using at most $latex p $ pours from containers of sizes in $latex S$

For example, we saw above that $latex {\bf F}(4,9;10) = (0,1, … , 13).$ But is this maximal for two containers and 10 pours? If we only allow 5 pours, the set $latex R$ reduces to size 7; and if we consider the smallest result not attainable, we get 2 quarts. That is very small! I’m tempted to use $latex \lfloor {\bf F} \rfloor$ to denote the smallest unattainable positive integer amount, but that’s only a whim.

Ultimately, this leads to two broad, open (as far as I know) questions:

Questions
What sets $latex S $ maximize $latex |{\bf F}(S;p)| $ for various $latex p$?
What sets $latex S $ maximize $latex \lfloor {\bf F}(S; p) \rfloor $ for various $latex p$?

Perhaps these have already been explored? I don’t even know what they might be called.

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