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Looks like it's called "Demi-chess" according to https://en.wikipedia.org/wiki/Minichess which describes several small board variants.
Demi-chess is a variant that is set up on a half of a standard chess board, that is, a 4x8 set-up. The "half-board chess" set-up with all its variants I called "Semi-chess" at the beginning of this forum, and at this point, I do not know if these are original terms that I coined or if these have been thought of by someone before me. So on this forum, I am looking at all variants of "half-board chess".
One site for Demi-chess is:
Polgar recommends the QuickChess, which incorporates all six pieces. I see no reason to simply exclude one for training and then have to introduce it at a later time. Another concern is that of the teacher's skill. Some principles that apply in regular chess do not apply as much or at all in variants. Part of the reason for this skepticism is the lack of presented expert sources that have investigated the matter. I am not against using simplified methods to simplify the process that students need to follow. I simply ask, why do you believe this particular variant is superior?
@minnesotachesscoach: You have pertinent questions. For starters, of course simplifying is not the desired goal, as I see it, to make it easier for beginners. (Let him/her choose the level that he/she wants to play.) Rather simplification through Semi-chess variants is desired to both 1) teach particular skills and 2) to temporarily avoid the distractions of more complex positions in standard chess. In short: simplifying for better concentration and focus.
Also, teaching standard chess concurrently with "Semi-chess", for example in the same class period, could be very useful while excluding a piece like the queen in Semi-chess should not be much of an issue in this way of teaching. Semi-chess has been shown to be a valuable tool; and while it is not purported to be THE superior variant (which I never said), I imagine that some variants out there would be inferior. What Polgar or Carlsen recommend are some of the good ones.
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