I went through the nim games page on this site. I had a recollection of having gone through it before as well.

Well, I admit that the solution to the problem is mind blowing. Once you tell me to convert the number of counters in binary and then do the XOR to find column with odd numbers of 1's...its all downhill from then on.

But on both occasions when i went through this game, what took me off was the lack of explanation (or perhaps my inability to see if there is one) behind our conversion of number of counters in a pile to binary.

I mean this is not the obvious thing I would do. I would probably tinker around with some other approaches rather than converting the number of counters to binary. It'seems the last thing I would give a shot at.

Even though this conversion to binary ensures separation of safe and unsafe positions which keep on oscillating after every move, is there not any other way around to achieve the same? And again what exactly prompts one to try binary conversion to achieve this separation?

Kindly explain to me the motivation behind our conversion of number of counters in a pile to binary.

Thank You
-Akash Kumar

Einstein - You know, earlier my preference mathematics. Later on i changed it to physics

Pincare: And why is that?

Einstein: I could not tell important facts from non-important ones.

Poincare: Starne that you say so. But now that you mention it, earlier i cherished physics..Now i cherish mathematics.

Einstein: And why is that

You should probably try Plainim first

https://www.cut-the-knot.org/recurrence/Plainim.shtml

which is an explicit version of Nim in which you do not directly do any calculations.

Or try playing with two piles. What would be a strategy here?

>I mean this is not the obvious thing I would do. I would
>probably tinker around with some other approaches rather
>than converting the number of counters to binary.

That's absolutely fine. Where did your tinkering take you?

>Kindly explain to me the motivation behind our conversion of
>number of counters in a pile to binary.

Your assumption is that one can do better on a second try even if he tried to do his best on the first. The motivation comes from the game with two piles. If, after you make a move, the piles have equal number of items you are assured of being able to make another move regardless of what your opponent does.

In general, the purpose of your move is to leave a situation "even" in some sense, like two piles of equal length. XOR provides the measure of the evenness.