a weblog of wordplay by Eric Harshbarger
Scrabble SquaresOn 1 March 2006 officially sanctioned Scrabble tournaments in the United States will start using a new wordlist. It will be based off the recently released fourth edition of the Official Scrabble Players' Dictionary.
This may not sound like a big deal to laymen, but believe me, I know there are many expert Scrabble players who have spent the last several months memorizing all of the new words. Not too mention all of the debate over and analysis of the introduction of two particular two-letter words, QI and ZA; which some feel will unbalance the game greatly.
A whole history and discussion of the various Scrabble word sources is way beyond the scope of my article here (that history includes "official" dictionaries that are not used in tournaments, "naughty" words, misprints, and so on). Not having played competitive Scrabble now for about 18 months, I'm not even that interested in the switch to the new tournament wordlist.
However, the new words did lead me to a recent logological challenge. After reading about my interest in word squares, another Scrabble player, Jeff Myers, told me of something he'd been trying to solve for many years now: was there a way to arrange the 100 tiles of a Scrabble game such that 4 5x5 word squares are created? (The blanks could be designated as whatever letters necessary.)
While little progress had been made with the old tournament wordlist as the word source, Jeff was hopeful that the upcoming, new wordlist (with 289 five-letter words added) might yield a solution. He asked if I might want to take a stab at it myself.
Why yes, I would like to.
I also mentioned the problem to a couple of other wordplay freaks that I know, including Mike Keith. Soon we were deep in 5x5 word squares.
There is no shortage of 5x5s that can be produced with the acceptable wordlist, but finding 4 of them that, together, match the Scrabble tile distribution is quite a task, indeed. There are so many 5x5s, that trying to exhaustively combine any 4 of them would take much too long. The trick then is to try to narrow down the number of 5x5s you actually consider viable candidates for the final quartet of squares.
For example, the strategy I first employed was to only build 5x5 squares in which the distribution of each letter roughly equaled one-fourth the number of that letter in a Scrabble game (the actual formula I used was int(N/4)+1 -- so, for the letter "O", of which there are 8 in a game, I only wanted to consider squares that contained, at most, 3 of them).
The blanks, of course, allowed for some leeway, but my search with this first restriction was unsuccessful. No solutions were found. At the time I was about to leave on a trip overseas for two weeks, so I could not implement any new strategies before I departed.
Mike, working independently, made good use of that time, and, two days after I found myself in London, he sent me an email pointing to this quartet of 5x5s. He had done it.
And, over the next couple of days he found over one hundred more solutions (including some which only use words from the old tournament wordlist).
Ours restricted searches were not exhaustive strategies, just plans to try to find some quartets of squares. None of us is sure how many solutions there actually are to this problem.
So, that's that.
Well, not quite. I couldn't resist using this knowledge to cook up a mischievous item. You see, in tournament Scrabble play it is common and completely within the rules to "track tiles" during a game. This means that players will keep track of which tiles are on the board by crossing them off their score sheets. Tracking tiles helps one know what tiles have not been played and (especially during a tight endgame) this can make decisions about future plays wiser.
The Official Tournament Rules explicitly state that "you may construct and use your own letter lists." I've seen players group letters with all of the vowels in a bunch and the consonants elsewhere. The high scoring tiles: J, K, Q, X, and Z, are often singled out somewhere on the score sheet. I always used a simple alphabetical listing which allowed me to speedily cross off most of the tiles at the end of a close game in about 2 minutes.
What I had never seen, however, was a "listing" of letters/tiles which took advantage of this new 5x5 word square knowledge. Players, evidently, are allowed to arrange the letters however they wish on their score sheet. So... why not arrange them in the quartet of 5x5 word squares shown above?
Why do this? Well, because those 4 squares list 40 valid five-letter Scrabble words. Maybe a player has a hard time remembering whether AZON takes a trailing S or not. Now, looking at his score sheet, he knows, "why yes, it does."
I know, I know, this probably violates some "spirit of the game" clause in the official rules. It is rather underhanded. But this article is just for fun and, er, well... fun and games. And in that spirit I'll provide a link to a convenient PDF-format score sheet with exactly that tile-tracking arrangement.
I'm not going to make a version of the score sheet for all 100+ quartets found. If some devious tournament player wants a different set of 5x5s, he will have to make another score sheet himself.
[25 February 2006]
Comments about this article:Wow - good job, Eric! VERY creative!
Posted by: Susi Tiekert
Posted by: Simon Lengyel
Posted by: jxano
The reason for having it do so was that we were looking for 10x10 squares, and the probability of finding one was low enough already without requiring the vertical words to be different.
Posted by: Graham Toal
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