Consider a deck of 52 cards in four mathy suits --two red: "adds" ($+$) and "subs" ($-$); and two black: "muls" ($\times$) and "divs" ($\div$)-- and thirteen numerical values --"A" (1), 2, 3, 4, 5, 6, 7, 8, 9, 10, "J" (11), "Q" (12), "K" (13).
A solitaire game on this deck arranges the cards into four stacks of thirteen cards each, according to two familiar rules:
- Cards in a stack must alternate color
- (Reading from the bottom of the stack) The cards must decrease in numerical value, although "K" is allowed to be placed on top of "A". Values within a stack must be consecutive.
For example, this is (part of) a valid stack of cards:
$$[bottom] \;\; [5-] \;\; [4\times] \;\; [3+] \;\; [2\div] \;\; [A+] \;\; [K\times] \;\; [Q-] \;\; [J\times] \;\; [10+] \;\; [9\div] \;\; \cdots \;\; [top]$$
Run-of-the-mill stuff, no? Well, consider that we assign a score to every stack of cards:
- With an initial value of zero, (and again reading from the bottom) we apply the arithmetic operation represented by each card in turn. (That is, you can think of each card as Reverse Polish Notation instructions.)
- Division rounds toward zero (so that every computation step yields an integer)
Edit. I changed the example to make more explicit use of rounding in the middle of a stack store computation, and to remove possibly-deceptive patterns in the suits.
The example partial stack would have a score of $-124$:
$$(0) [5-] \rightarrow -5$$ $$(-5) [4\times] \rightarrow -20$$ $$(-20)[3+] \rightarrow -17$$ $$(-17)[2\div]\rightarrow -8.5 \rightarrow -8$$ $$(-8)[A+] \rightarrow -7$$ $$(-7) [K\times] \rightarrow -91$$ $$(-91) [Q-] \rightarrow -103$$ $$(-103) [J\times] \rightarrow -1133$$ $$(-1133) [10+] \rightarrow -1123$$ $$(-1123) [9\div] \rightarrow -124.777\dots \rightarrow -124$$
Finally, we compute the game score:
- The game score is the absolute value of the sum of the stack scores (NOT the sum of the absolute values).
Question: What's the largest possible game score?
A fairly naive computer search gave me a value of $915,382$. I'll accept an answer that proves this (or whatever better number) is optimal. (Proof can come in the form of a comprehensive computer search algorithm, though I'd prefer something "logical".)
UPDATE. I conducted a new and (I believe) comprehensive computer search, and have confirmed the $915,382$ maximum. (Perhaps-unsurprisingly, maximal arrangements --for the most part-- put "$+$" and "$\times$" cards into two stacks, and "$-$" and "$\div$" into the remaining stacks. There can be some fiddling with black bottom cards, since they all provide the same result when their "operation" is performed on zero.) @FelixCQ's observation that stack-starting values come in pairs of opposite color was the key to constructing a manageable search strategy, so I'm inclined to accept his answer. However, I'll leave the question open for a while longer in case anyone wants to put forth an analysis that doesn't involve checking 5.67 billion individual cases.