All or Nothing Trickle Treat

Just in time for election season, we present a ternary trickle down principle. First, we note that Martin Gardner (1914-2010), the best friend mathematics ever had, would have turned 98 on 21 October. There is a Twitter feed in his honor at WWMGT, for which suggested contributions (What Would Martin Gardner Tweet?) are welcome. 

In the USA, Gardner did for recreational mathematics what Julia Child did for recreational cooking. His influence spread much further afield actually, and like Child, he also inspired quite a few professionals along the way: mathematician and card expert Persi Diaconis has correctly noted, "he has turned thousands of children into mathematicians," playfully adding, "and thousands of mathematicians into children." After the above culinary comparison was made, author and Gardner friend Chris Morgan commented, "I was lucky to have spent time with Julia in the 1980s (I took three cooking classes with her in Boston). She was very much like Martin in many ways." 

Last October here, we made available here numerous card classics that Martin had published in his legendary "Mathematical Games" columns in Scientific American, following up on the earlier June 2010 and August 2010 Card Colms. 

October is now well established worldwide as the anchor month for annual Celebration of Mind events, which promote Martin's wide interests and remarkable and extensive written legacy. These gatherings bring together young and old, trained and untrained, to have fun with logic, puzzles, magic, optical illusions and more, very much in the spirit of the ever-curious Gardner. 

This year, in addition to seventy plus Celebration of Mind events listed at the event map here, there are an equal number of Flexagon Parties going on all over the globe, inspired by Vi Hart's viral videos on YouTube (don't miss the fourth food themed one, which would have delighted both Child and Gardner). Vi was of course inspired by the Hexaflexagons article which marked the start of Gardner's quarter century tenure atScientific American. 

Incidentally, it's not too late to host or attend events of either type. For instance, this year's MAA Celebration of Mind event is set for 5th December. Please consider joining or hosting an event, formal or informal, wherever you are. 

October is also when the world's most extensive and successful mathematics outreach programme Maths Week Ireland is held. This year, over a nine day period ending on Martin's birthday, it exposed over 150,000 people of various ages in both the Republic of Ireland and Northern Ireland to the joys of mathematics and logical thinking, with several nods to Gardner thrown in for good measure. The closing event was really a Celebration of Mind show and tell for the whole family, facilitated by a dozen presenters, which over a thousand people attended. I had the pleasure of participating, sometimes alongside notable Gathering for Gardner regulars such as Fernando Blasco. Top notch speakers included Colin Wright and David Singmaster. 

It was while driving towards Sligo a fortnight ago with Maths Week Ireland mainstay Dr. Maths, on the way to address hundreds of school children, that I was introduced to a delightful recreation with triangles which forms the basis for this month's offering here. Like the title below, it's hard to resist, and I'm confident that Martin would have found it fascinating.

Humble Contribution

Start with a row of n items randomly selected from three possible types, say red, blue and white poker chips. Underneath each pair in the row, place a third item to form triangles, according to this rule: 


If the two vertices above are the same colour, then the third one matches both, whereas if the two vertices above are different colours, then the third one is different from either. 
In other words, in each such downward-pointing tiny triangle, one never has only two vertices matching. When it comes to vertex matching, it's an all or nothing situation. 

Once the initial row of n items is processed to yield a second row of n - 1 items below it, process that row in the same way, to get a third row of n - 2 items underneath it. Continue, eventually getting a single item at the very bottom of an n-row triangle. 

Here are three examples with poker chips: 

Note that in each three-chip downward-pointing triangle—but not necessarily in their upward-pointing cousins—any two vertices determine the third according to the All or Nothing rule. Hence, upon reflection, or should we say upon rotation, any completed triangle (of any size) may be turned 120 degrees in either direction to yield an equally valid arrangement. Indeed, the second and third images above are exactly such rotations of the first one! 

Here is the main question we wish to ponder: 


Given the top row of such a triangle, of any size, can one easily predict what the final bottom vertex will be?
Note the pleasing random-seeming arrangements of "solid sub-triangles" of various sizes in red, blue and white, which make up the larger triangles above. It almost suggests a fractal behaviour. 

This can all be implemented with cards, for instance, using only three suits (which coincidentally was considered in the last Card Colm). 

Alternatively, we could use face-up red and black and face-down cards for the three types, as illustrated below. 

The secret here is to focus on the fourth row from the bottom: specifically its starting and finishing items. Those two cards alone determine the final bottom card according to the All or Nothing rule! No matter what other two cards are between the face-down blue-backed card and the 7 of Hearts, the final card will always be Black. In fact for every downward pointing "four triangle," consisting of ten cards, any two of the three vertex cards determine the third according to the same rule. 

What's so special about four? Here's a hint: it's also true for downward pointing "ten triangles" consisting of 45 cards headed by a row of ten cards. Here's a perfectly reasonable question to also ponder: what is the next number after ten which also works? (You'll need eight decks of cards.) 

Now suppose the following five cards are laid out, and our goal it to speedily guess the nature of the last card if the associated fifteen card triangle is completed. 

It's easy to see that the first card in the next row should be Black. Likewise the last card in this row should be Black, so the final card of the completed triangle should also be Black. 

A similar "trickle down" approach can be used to predict the nature of the final card starting with rows of length six or seven. For longer rows, it may be easier to go the other direction, noting that, 
Given a triangle of any size, it can be extended upwards in exactly 3 distinct ways, by first deciding on the nature of any single card in the row above, and then filling in the rest to be consistent with the All or Nothing rule. Just be sure to always "trickle down" (not up)!
It's not hard to do this for a row of nine cards, extending it mentally to a row of ten and then quickly deducing the nature of the final card at the bottom if eight more rows were constructed. 

The principle underlying the above observations may be found in the upcoming paper "Triangle Mysteries" by Erhard Behrends & Steve Humble, which is due to appear in The Mathematical Intelligencer in June 2013. There, the result hinted at above is generalized, and the connection with Pascal's triangle is explored in depth. Thanks to Steve (Dr. Maths) for the preview! 

Note that from a large downward-pointing triangle—which is as we noted completely determined by any of its three sides—one can also carve out Halloween hexagons with interesting properties. Have fun!