The Linear Frog

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The Linear Frog

I was given a problem recently, and I was really pleased to get the solution. I wonder if you can (a) solve it, and (b) tell me where it comes from. I've not been able to track it down at all.

 I'm still gestating the post about Puzzle World vs Crypto World. Not long now. I hope.
So here it is. I'm going to present it in two forms, to account for the difference between Puzzle World and Crypto World.

The "Puzzle World" version ...

There's a set of lilypads stretching off beyond where the eye can see, and we number them with the non-negative integers. So we label them, starting with 0.

There's a frog who, one day, for no obvious reason, decides to hop on the pads, starting somewhere random, and then hopping the same distance each time, off into the distance. No, I have no idea why, but it seems that it just wants to keep going.

But its plans were overheard by an unnamed predator. Our predator is very, very short-sighted, and when close to one lilypad is unable to see any others, but it's lightning fast. Sort of. So it decides to visit lilypads, one at a time, in an attempt to catch the frog. There's no limit to how far the predator can travel between lilypads, but it needs to spend a unit of time at each one to detect whether or not the frog is there.

So for every time step the predator moves to a lilypad, and the frog jumps to the next in their sequence.

The question is:

• Can the predator catch the frog?

The "Crypto world" version

So here is the formulation in more precise and strictly mathematical terms.

• Can you find a sequence $s(t)$ such that for every arithmetic sequence $a+dt$ there is a $t$ with $s(t)=a+dt$?

We are, of course, working in integers, and assume that $a$ and $d$ are non-negative.

Extension

Once you solve the problem for the frog moving in a linear pattern, what if it moves in a Cubic? Or Quintic? Or Exponential? What if $a$ and $d$ are allowed to be any integer (breaking the original idea of the lilypads going off in one direction).

Problem source ...

So over to you. Firstly, can such a sequence be found? If so, what is it, and if not, can you prove it?

Secondly, do you know a source for the problem? If so, let me know.

Enjoy!

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