Puzzle time, count the hydras
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This one came up at work recently. AI will make short work of this sort of thing eventually, but as it stands, it fell to yours truly. I'm only asked if the on-site PhDs can't figure something out.
A hydra is an animal with one or more heads, where each head must have a different color. For generality, let's give each color a number. So there is color 1, color 2, and so on.
These hydras are immigrating into your country at a certain rate, with each type of hydra having its own rate. Hydras with a single head of color 1, call them 1-hydras, come in at an unknown average rate of x per hour. 2-hydras come in at some other rate of y per hour. 1,2-hydras, which are two-headed hydras with head colors 1 and 2, come in at their own rate of z per hour.
You have a gate through which all hydras enter, but the information tracking at this gate is primitive. Every minute, the gate reports back whether it's seen at least one head of color 1, and whether it's seen at least one head of color 2, within the past minute. These heads could be attached to 1-hydras, 2-hydras, or 1,2-hydras. The gate doesn't tell you that.
How do you deduce, from the information at the gate, collected over an arbitrarily long time span, x, y, and z?
This has to do with the poisson distribution, which is unfair to assume anybody could figure out for themselves, so I'll just lay that part out: if you know the fraction of 'positive' minutes for a given head color, in other words the fraction of gate readings where that head color was seen at least once, then you also know the overall rate per minute, on average, of that head color passing through the gate. This remarkable formula is -ln(fractionOfNegatives).
So if you see at least one color 1 head 25% of time, then color 1 heads are passing through the gates at a rate of -ln(.75) per minute, or .287 heads per minute.
Then for extra credit, extend the solution to hydra species of any number of heads. Such as, imagine the gate now keeps track of three head colors, and that there are hydras of 1, 2, or 3 heads, and you want to count the species of each, coming through your gate.
Well, writing it out, it's not quite as elegant and presentable as a puzzle as I'd hoped. Oh well.
@Horace said in Puzzle time, count the hydras:
How do you deduce, from the information at the gate, collected over an arbitrarily long time span, x, y, and z?
Attach a color CCTV camera???
(Actually, I have no idea and no really sure where to start. I will think about it, but doubt I will be able to do the solution.)
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Nice framing of the problem, @Horace.
Kindly clarify this ... you say the "gate" will report whether it has seen "at least one" head of a certain color in the last minute. Does that mean, the gate is reporting only "one bit" of information for each color per minute (as opposed to reporting the number of heads for each color it has seen in the last minute)?
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Nice framing of the problem, @Horace.
Kindly clarify this ... you say the "gate" will report whether it has seen "at least one" head of a certain color in the last minute. Does that mean, the gate is reporting only "one bit" of information for each color per minute (as opposed to reporting the number of heads for each color it has seen in the last minute)?
@Axtremus said in Puzzle time, count the hydras:
Nice framing of the problem, @Horace.
Kindly clarify this ... you say the "gate" will report whether it has seen "at least one" head of a certain color in the last minute. Does that mean, the gate is reporting only "one bit" of information for each color per minute (as opposed to reporting the number of heads for each color it has seen in the last minute)?
Right. You get a binary for each head color, per minute. Either zero of that color passed through, or some positive number. But you don’t know how many.
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This one came up at work recently. AI will make short work of this sort of thing eventually, but as it stands, it fell to yours truly. I'm only asked if the on-site PhDs can't figure something out.
A hydra is an animal with one or more heads, where each head must have a different color. For generality, let's give each color a number. So there is color 1, color 2, and so on.
These hydras are immigrating into your country at a certain rate, with each type of hydra having its own rate. Hydras with a single head of color 1, call them 1-hydras, come in at an unknown average rate of x per hour. 2-hydras come in at some other rate of y per hour. 1,2-hydras, which are two-headed hydras with head colors 1 and 2, come in at their own rate of z per hour.
You have a gate through which all hydras enter, but the information tracking at this gate is primitive. Every minute, the gate reports back whether it's seen at least one head of color 1, and whether it's seen at least one head of color 2, within the past minute. These heads could be attached to 1-hydras, 2-hydras, or 1,2-hydras. The gate doesn't tell you that.
How do you deduce, from the information at the gate, collected over an arbitrarily long time span, x, y, and z?
This has to do with the poisson distribution, which is unfair to assume anybody could figure out for themselves, so I'll just lay that part out: if you know the fraction of 'positive' minutes for a given head color, in other words the fraction of gate readings where that head color was seen at least once, then you also know the overall rate per minute, on average, of that head color passing through the gate. This remarkable formula is -ln(fractionOfNegatives).
So if you see at least one color 1 head 25% of time, then color 1 heads are passing through the gates at a rate of -ln(.75) per minute, or .287 heads per minute.
Then for extra credit, extend the solution to hydra species of any number of heads. Such as, imagine the gate now keeps track of three head colors, and that there are hydras of 1, 2, or 3 heads, and you want to count the species of each, coming through your gate.
Well, writing it out, it's not quite as elegant and presentable as a puzzle as I'd hoped. Oh well.
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@Horace said in Puzzle time, count the hydras:
How do you deduce, from the information at the gate, collected over an arbitrarily long time span, x, y, and z?
How do you deduce what exactly?
@Klaus said in Puzzle time, count the hydras:
@Horace said in Puzzle time, count the hydras:
How do you deduce, from the information at the gate, collected over an arbitrarily long time span, x, y, and z?
How do you deduce what exactly?
x y and z were defined in the problem. They are the rate of entry of 1-hydras, 2-hydras, and 1,2-hydras.
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We're actually pursuing a patent for this. One cannot patent a mathematical process, but our patent lawyers contextualize it into our physical instrumentation. They have some experience with this. They expect it'll get rejected at first pass, but maybe approved on appeal.
Maybe in a couple years I'll update this thread with the good news of a patent. But I would bet against it. -
We're actually pursuing a patent for this. One cannot patent a mathematical process, but our patent lawyers contextualize it into our physical instrumentation. They have some experience with this. They expect it'll get rejected at first pass, but maybe approved on appeal.
Maybe in a couple years I'll update this thread with the good news of a patent. But I would bet against it.@Horace said in Puzzle time, count the hydras:
We're actually pursuing a patent for this. One cannot patent a mathematical process, but our patent lawyers contextualize it into our physical instrumentation. They have some experience with this. They expect it'll get rejected at first pass, but maybe approved on appeal.
That makes sense. When you think about it, Google’s “search” algorithm is a mathematical process too, yet you can bet Google has all sorts of patents around it. It may take a few years to prosecute the patent, but congratulations on getting the patent application filed.
