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# he third project is a combinatoric problem concerning the words given in sgb words.txt

INSTRUCTIONS TO CANDIDATES

Project Three: Packing Words in Bins

The Problem To Solve

The third project is a combinatorial problem concerning the words given in sgb-words.txt that contains 5757 five-letter English words, sorted by their frequency of actual use.

Given a list of suitably randomly chosen words from this dataset, (the automated tester will choose these words for each run) for example

upset, abaci, ahold, runty, bones, kulak, mange, jerky, crook, frisk, pulse, frump, harpy, paste, poles, knows, musks, ensue, wahoo, gouts, elide, paced, terns, hewer, fault, stave, train, yukky, fluky, leash, chuff, fined, glyph, enema, shrub, blend, chant, fungo, genii, midst, dolly, deism, grook, couch, spina, torsi, estop, mylar, rigid, ichor

your task is to pack these words into separate "bins" under the constraint that no two words in the same bin may have the same letter in the same position anywhere. For example, the words frump and gouts cannot go in the same bin, since both have the same letter u in the middle position. Under this constraint, the previous fifty words can be packed into seven separate bins using, for example, the following arrangement, one of the exponentially many equally valid arrangements possible. (Packing of these words into only six bins turns out to be impossible within the constraints of this problem.)

Bin 1: [bones, fluky, paste, leash, kulak, ahold, estop, spina] Bin 2: [pulse, fined, gouts, jerky, crook, abaci, enema] Bin 3: [poles, fault, runty, deism, grook, ensue, blend] Bin 4: [mange, frisk, terns, dolly, upset, chuff, rigid, glyph, ichor] Bin 5: [fungo, paced, torsi, elide, chant, knows, mylar] Bin 6: [musks, frump, harpy, genii, stave] Bin 7: [couch, yukky, hewer, midst, wahoo, train, shrub]

Only the total number of bins used to achieve the packing arrangement matters; how evenly the words are distributed into these bins is irrelevant. (In practice, minimizing the number of bins tends to create roughly equally sized bins.)

An equivalent but more computer science-y way to phrase this problem would be to ask for a minimum vertex colouring in the SGB word graph where two word vertices are connected by an edge if and only if those words share the same letter in at least one position.

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