Mathrix

Description

Mathrix is an abstract piecepack solitaire puzzle for the mathematically inclined. The goal is to remove all but one coin by formulating mathematical equations using strings of coins.

(Random) setup for a game of Mathrix

Rules

• https://ludism.org/piecepack/org/rules/Mathrix.pdf (local copy)
• https://web.archive.org/web/2016/http://www.piecepack.org/rules/Mathrix.pdf (original copy)
• MathrixTextVersion

Reviews & Comments

Mathrix was an entry in the SolitaryConfinement design contest. -- ClarkRodeffer

Here is some post-contest feedback that ClarkRodeffer received from Phillip Lerche:

"Another well-written ruleset of a very clever game (in the intellectual sense) that I found a touch dry. It struck me that you didn't really need the grid - you could just as easily have laid the rows of coins out without it. So I felt there wasn't much use of the piecepack as such (you could as easily have used playing cards).

"Neat that you came up with 2 such different games - the theme-dripping PPP and the clean math abstract!" -- added by ClarkRodeffer with Phillip Lerche's permission

Version 1.0.0 includes some minor rule and wording clarifications based upon player comments. -- ClarkRodeffer

Mathrix is certainly fun. It is refreshing, a mind-stretcher rather than a mind-bender. (Marty calls it "brain yoga".) The rules are easy to understand and remember, but interesting strategies and tactics emerge from them -- my favourite kind of game. Understanding the relationships among small integers is key here. I would recommend Mathrix for anyone who likes puzzles, wants a math refresher, or wishes to teach math to kids.

I have a reservation about the game that is itself a puzzle of sorts. It seems that some numbers in Mathrix can be treated as completely identical to other numbers by using unary operators. For example, 4 can always be treated as equivalent to 2 when necessary, because sqrt(4)=2. Similarly, a null coin can always be treated as an ace coin as well, because cos(0)=1. It seems to me that a 6x6 "mathrix" of such equations (leaving out the cases 0=0, 1=1, etc., thus 30 equations in all) would render Mathrix trivial, since you could always remove any coin from the board. Can such a table be constructed? How abstruse would the math have to be to show 4 can be converted to 5? I'm interested to hear opinions on this. Barring the creation of such a table, or even most of one, by piecepacking spoilsports :) , Mathrix is plenty fun.

In sum, a fine game at root, Clark, modulo my reservations about operations that would subtract from the game. More power to you.

--RonHaleEvans

After I wrote the review above, Mark A. Biggar pointed out that you really only need 15 equations, "as you only need to convert each pair one way". He suggested the following table as a way to "break" Mathrix:

ln(1) = 0
floor(ln(2)) = 0
floor(ln(ln(3))) = 0
floor(ln(sqrt(4))) = 0
floor(ln(floor(sqrt(5))) = 0

floor(sqrt(2)) = 1
floor(sqrt(3)) = 1
floor(ln(4)) = 1
floor(ln(5)) = 1

ceil(sqrt(3)) = 2
sqrt(4) = 2
floor(sqrt(5)) = 2

ceil(sqrt(sqrt(exp(4)))) = 3
ceil(sqrt(5)) = 3

sqrt(4) = floor(sqrt(5))

Perhaps the floor and ceiling operators should be outlawed in the next version of Mathrix, although that seems very ad hoc and against the spirit of the game.

--RonHaleEvans

I don't think floor and ceiling (or round) should be outlawed, because like Ron said, it's against the spirit of the game. However, purists may want to avoid the use of sqrt (because technically, it requires the use of an additional two), ln and exp (because they require the use of the number e, and likewise for logarithms in other bases), factorials and gamma functions (because they require the use of all of the positive integers less than the argument), and so on and so forth. The main rule to keep in mind is clearly stated in the ruleset: "If it feels like cheating, it probably is." --ClarkRodeffer

That same reasoning would also outlaw cosh(x), sinh(x) and tanh(x) for the same reason exp(x) is. cosh(x) = (exp(x) + exp(-x))/2, sinh(x) = (exp(x) - exp(-x))/2 and tanh(x) = sinh(x)/cosh(x). And we also can't use .1, .2, .3, .4 and .5 because of the implied x/10 in there. So an ultra-purist version only allows the use of the integers 1, 2, 3, 4, 5 in formulas. Of course I'm taking this as a challenge and hope to create a new table that lives by all the above restrictions. -- Mark A. Biggar

Excellent points, Mark! I know you're up to the challenge. However, the decimal point is a gray area for me; I'm not so sure that placing a decimal point between two integers should be disallowed. For example, 3 / 2 = 1.5 is certainly in the spirit of the game. :-) -- ClarkRodeffer

I just noticed that floor(abs(sin(X))) = 0 for all integer X (considered as radians), therefore any pair of values (A,B) the equation floor(abs(sin(A))) = floor(abs(sin(B))) lets any two coins match and makes the game trivial. To save the game we should probably require that any equation used be of the form "A = ..." where A is a coin. I'm still working on a cheat list with this restriction. -- Mark A. Biggar

Again, an excellent point. But what about possibly restricting both sides of each equation to rational expressions? -- ClarkRodeffer

I believe that is an equivalent condition to mine -- Mark A. Biggar

Mathrix feels more like an open design than a tightly constructed game: the creativity (i.e. the decisions) in the game lie in the choice of operators between numbers. The very open-ended rules here allow a lot of tinkering:

• Can operators apply to just one number (square root, etc.) or must they indicate operations involving two separate numbers?
• Are rational numbers involved or just whole numbers?
• Can the distributive property be applied to multiply several consecutive numbers times zero or one?
• Does "No gaps" mean that the gaps are ignored or that gaps are not allowed for building equations?
• Do all coins need to be in a straight line or is it just enough for each of them to be orthogonally adjacent to the next coin?

(...)

As a consequence of this, Mathrix will provide sure entertainment to Mathematics aficionados, whereas casual players might not enjoy the grey areas that the rules deliberately leave open.

Antonio Recuenco Muñoz

CategoryGame DisplaceDifferencedPiecesCommonOwnershipCategory SolitairePuzzlesCategory