Nimstring position construction questions

[amdw]

• Is it possible to construct a position of value *0 which is a first-player win in strings-and-coins?
• Is it possible to construct a position of positive Nimber value (e.g. *1) which is a second-player win in strings-and-coins?
• How can positions of arbitrarily high Nimber be constructed?

[amdw]

It is quite simple to construct a position of Nimstring value *0 which is a first-player win in strings-and-coins. In fact, it is simple to make the strings-and-coins minimax value of any position arbitrarily large and positive without affecting its Nimstring (CGT) value.

Every independent capturable coin (connected to the ground) has Nimstring value *0, but adds 1 to the strings-and-coins minimax value, as it gives the first player a free point. So this is a (fairly trivial) way to answer the first question.

If there exists a position of value *0 without any capturable coins which is a first-player win, then the winning move must result in a position of positive Nimber value (as opposed to a loony move), as loony positions always have non-negative strings-and-coins minimax value.

[amdw]

On the second question, a trivial example of a position of value *1 which is a second-player win is a single independent 2-chain, which has minimax value -2.