a polynomial problem

The latest problem from fakalin. Took some wrong turns and hints to solve it…

Given a polynomial \(f: \mathbb{Z}\to \mathbb{Z}\) with positive integer coefficients, how many evaluations of \(f\) does it take to obtain the polynomial?

(An \(f: \mathbb{R}\to \mathbb{R}\) polynomial with real coefficients would take the number of degrees plus 1 to specify, which, if it held in this case, would render the answer unbounded. But the correct answer in this case is surprisingly small.)
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How do escalators work? I’ve wondered for years how escalators recycled their step blocks internally. At one point I thought they slid past each other on all four faces to save on turning radius (because I thought the blocks locked along grooves). Today I see an escalator under repair. Now the answer is clear. It’s much simpler than that: the blocks just turn along a track in the most obvious way imaginable.

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a card problem

Here is a problem quoted from fakalin. A full deck of cards has 52 cards. Suppose 10 of them were face up and 42 were face down. You are in a dark room holding the deck. How do you rearrange the deck into two subdecks so that they have the same number of cards facing up?
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can you tell Asians apart?

There is a Chinese, a Japanese, and a Korean in here. Identify them.

As for the answer:
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tax forms must be designed by idiots

CA income tax form is the worst.
MA is only slightly better.
The federal one is a disaster but at least I’m used to it.
These things require reverse-engineering the spagetti code behind the instructions in order to see the actual calculations, which are all fairly simple. And yet, there is no logic to the instructions, like why the apportioning of income for non-residents need to be calculated multiple times, or why rate and value calculations are interleaved in random order, or why two forms that should give the same answer, don’t… Argh!