# A balanced banquet

As your king's fitness adviser you have been tasked with optimising His Majesty's diet. This entails making sure that the king eats equal amounts of carbs (rooks), fat (bishops) and protein (knights). As the king enjoys his meals you better have enough of each on the table.

Task: Place a white king and equal numbers of black rooks, bishops and knights on a chess board such that the following becomes possible:

1. Only white moves.
2. White must capture a black piece with every move.
3. White must capture all black pieces.
4. White must never be in check except possibly before their first move.

The goal is placing as many black pieces as possible subject to the stated rules.

• We've seen several puzzles recently with a goal (roughly) of "find the best/most ...". As a reminder, this community decided to do away with open-ended puzzles. For an actual optimization problem, rather than a "best score wins" game, "answers should come with justification of why they are optimal; an answer without this is not a full answer." If an answer cannot meaningfully be expected to show reasoning as to why it is indeed THE best answer, the question is likely improper.
– Rubio
Commented Apr 9, 2023 at 17:36
• I was led to believe that these are not rigorously enforced rules, especially in chess (see for example comments by not-exactly-firebrand bobble here: puzzling.stackexchange.com/q/120364/73836). If the optimization tag is what bothers you I'll be happy to lose it. Besides, an optimal solution trivially exists and proving optimality may be difficult but is certainly feasible (I do not understand the meaning of "meaningfully expect"). Commented Apr 9, 2023 at 19:04
• @WeatherVane No, there cannot be 64 filled squares. Observe that if in $i^{th}$ move the white king captures piece $P_i$ and in $j^{th}$ move ($j>i$) king captures piece $P_j$, then initially, $P_j$ MUST not attack $P_i$, else $P_i$ cannot be captured by king in the first place since it was supported by $P_j$. If there were 64 filled squares, then $P_{63}$ will always "support" some earlier captured piece. So, this is not possible. Moreover, its not quite hard to see that there can be maximum 22 pieces initially. I believe this can be reduced to 16, but I don't have time now. Good Luck :) Commented Apr 9, 2023 at 20:01
• C'mon, @Rubio, just take a step back and compare your interpretation with what she has actually written. They don't really match, do they? Anyway, as I said I do not only suspect there is an optimal solution, I know it for a fact. So I take it that is settled, then. Commented Apr 9, 2023 at 20:09
• @Rubio You are misquoting by omission, conveniently leaving out "It's not strictly required, especially for [chess] since there's a chess-loving lobby hereabouts :) but [...] it would help. The idea is to [...]". Commented Apr 11, 2023 at 5:03