Spiked Math Games  // Math Fail Blog  // Gauss Facts  // Spiked Math Comics

# fact-003

MFT - The Elements Game - February 21, 2012
• Currently 4.6/5
• 1
• 2
• 3
• 4
• 5
Made it just in time today (10:40pm my time)!

Today, we're discussing Sheldon's car game with penny.

Hopefully someone out there is able to solve GG for the elements digraph (or find a reference to someone who already has)! Does there exist a player 1 winning strategy? Player 2? We should all be using the same set of elements though... hmmm...

Yes, there does exist a winning strategy for Player 1.

Consider the 6 elements beginning with M. All but two end with M. The others end with E (Manganese) and Y (Mercury). Both elements starting with E and both elements starting with Y end with M. So once an element beginning with M is reached, there are exactly 6 + 2 = 8 moves remaining, eka-thorium notwithstanding. Thus Player 1 wants to start with an element ending with M, such as, say, Helium.

That was fast!

I don't get, why there should be only 2 elements not ending with m. When i read your post i just thought of these: iron, oxigen, copper, gold, lead. So i would guess the problem is much harder to solve.

Iron, oxygen, copper, gold, and lead don't end with 'M', true, but they don't begin with 'M', as sean mentioned. The 6 elements which begin with M are: Magnesium, Manganese, Meitnerium, Mendelevium, Mercury, and Molybdenum. Of these, only Manganese and Mercury don't end in "M", and thus, by the rules of the game, can take you out of that set.

There are three elements beginning with E (Europium, Einsteinium, and Erbium, all of which end in M. That means that, in the game, Manganese must be followed by one of these three, and then by another M element. Effectively, the Manganese-E*m pair acts as a pair of moves which puts one back at needing an M.

Similarly, there are two Y elements (Yttrium, Ytterbium), both of which also end in M, making the Mercury-Y* also force you back to needing another M.

So any sequence which begins with M is going to continue requiring M elements (with one E element and one Y element in the sequence) until all M elements are used up. There are 6 M elements, so whoever is forced to pick a 7th M element loses. Adding in the E and Y element, that gives 8 "safe" moves before needing the 7th element. Since players alternate, whoever chooses the first M element will be forced to choose the 7th M element, and thus lose.

So the first player can win by starting with an element, any element, which ends in "M" but does not begin with M.

What might be more challenging is the corresponding game using element symbols instead of names. Helium->Einsteinium->Tin->Nickel->Indium->Niobium->Bohrium->Hydrogen->Hafnium->Fermium->Magnesium->Gallium->Gold->Unnilhexium->Mercury->Gadolinium->etc, etc, etc

So it sounds like the writers of the show had already calculated the winning strategy.

There aren't that many "ending" letters in the periodic table: elements only end with: M, N, E, C (Zinc), H (Bismuth), D (Lead), Y, S (Phosphorus), or R (Sulfur). That's only 9/26 of the alphabet!

Oops, forgot L (nickel) and T (cobalt). So 11/26

I was hoping that I could find a series that didn't allow the choice of an element ending in m. At least from my brief glance it doesn't appear so. I suspect you could chose in such a manor that would force the m's to be used up. I'm enjoying the new schedule!

iron nitrogen neon nickel lead...then its on to elements ending with m

1. It seams to me that in English this game is a bit pointless (unless you want short games, not actually allowing to use the full knowledge of the periodic table), since a lot of elements' names end with 'm' and few begin with it.

2. I'm sure most of you know Tom Lehrer, but since we're on the subject: http://www.youtube.com/watch?v=DYW50F42ss8&feature=related

3. Sine I'm on the subject of Tom Lehrer, for those mathematicians who don't know him yet (are there any?): http://www.youtube.com/watch?v=UQHaGhC7C2E

Maybe you could do a comic about him?

@Blaise Pascal: with the symbols, a win for the first player is to say "Tungsten". Its symbol is W, a single letter, and there are no other symbols beginning with W, so the first player has won.

Welll... what about either or? So Tungsten could be followed by something that began with a 'n' or a 'w' so Neon would work then the next element begins with 'n' or 'e' so einsteinium with 's' or 'm'. This would also block out the 'm' linkage @Pascal as words like molybdenum would also allow oxygen.

Giving this, there are 118 known elements as of 2011. I will get back on this later =) (unless someone already has a winning strategy)

After two hours of programming I can confirm, that there exists a winning strategy for the first player for every element that ends with an "M", and does not exists for any other :P
By the way the _complete_ game tree represented as strings is about 1.14GB ^^

It's impressive how people spend HOURS on solving some random problem they found ont he internet. Still, absolutely awesome.

(Note: You must have javascript enabled to leave comments, otherwise you will get a comment submission error.)
If you make a mistake or the comment doesn't show up properly, email me and I'll gladly fix it :-).

Welcome to Spiked Math!

Hello my fellow math geeks. My name is Mike and I am the creator of Spiked Math Comics, a math comic dedicated to humor, educate and entertain the geek in you. Beware though, there might be some math involved :D

New to Spiked Math?
View the top comics.

New Feature: Browse the archives in quick view! Choose from a black, white or grey background.

Other Math Comics