# On Hints in Puzzles

I played a game of Paperback with a couple of friends from Imperial yesterday. The game involves players making words from letter cards in their hands, and using them to buy additional, more powerful letter cards and/or cards worth victory points. It’s pretty much a deckbuilder crossed with a word game; while anagramming skills are indeed important, they aren’t always necessary (for instance, I managed to form the word WORK to buy the second highest tier victory point card) and certainly aren’t sufficient (keeping one’s deck efficient is certainly important).

In any case, the game reached a point where one of my classmates seemed to be agonising over what was presumably a tricky hand. The game does have a number of cards that use digraphs such as ES, TH or ED, which can make searching for words a little trickier. The common vowel (a vowel that players are all allowed to use in their words in addition to the cards in their hand) was an O, and the hand in question was [IT], [R], [B], [U] and two wilds. I immediately saw a solution when he sought help (BURrITO or BURrITOs; none of the cards had abilities dependent on word length) and let him know that a ‘perfect’ solution for his hand existed, though we decided to let him figure it out for himself. This started to drag on, though, and giving him the answer directly seemed excessive; I thus decided to drop hints that would hopefully lead him to the solution.

Technically, the search space that he would have had at this point would have been $5! + 6! \times 26 + {7 \choose 2} \times 5! \times 26^2$ which is about 1.7 million possibilities, though of course one does not simply iterate through them; combinations like OqRzBITU would probably not even be considered directly, especially since the distribution of letters in English certainly isn’t memoryless (the Q being a good example; it is rarely followed by any character other than U). I gave a few hints, in this order:

1. The word is a common word that a person in middle school would likely know.
2. The word can be made with just one of the blanks.
3. The blank you use matches one of the letters already on the table.
4. You can eat it.

We could argue that the point of a hint could be to reduce the search space being considered by disqualifying some solutions, and/or serve as a way to encourage the algorithm to explore cases more likely to be correct first. Interestingly, it seems that clue 1 was neither of the above, though. It didn’t really cut down the search space much, and as far as I know it’s not easy to use this information to organise one’s search. Its point was mainly to reassure my classmate that the solution wasn’t too esoteric and he would know it was a valid solution upon considering it. I didn’t think I had played any particularly uncommon words earlier in the game (the rarest might have just been TENSOR or something), though I think I might have played Anagrams with him before, where I know I’ve called out some more… interesting words.

The next two clues do actually substantially reduce the search space in a sense though; the second clue slashes it to just $6! \times 26 = 18720$ and the third further cuts it down to $6! \times 6 = 4320$. I would think that after the third hint, the search should be pretty fast; I would probably try doubling up on the R and T first as they’re common.

Nonetheless, hint number 4 seemed to be the hint he responded best to, and it falls into the second category (encouraging the search to explore cases more likely to be correct). I’m not sure how humans solve anagrams, especially when blanks are involved; I personally seem to look at letters, intuitively construct a word that looks like it incorporates the key letters present, and then verify that the word is constructible (in a sense, it is a generate-and-test approach). Given the point values of the tiles, I saw including the B, U and IT cards as being the most important; thinking of words that use these patterns resulted in BITUMEN and BURRITO fairly quickly, the latter of which was feasible.

I noticed that I tend to bias towards using the constraint letters near the beginning of the word though; this might be a cognitive issue with trying to satisfy the constraints quickly, in an unusual form of the human brain often being weak at long-term planning. This wasn’t really the case with the burrito example, but another friend had a hand of ten cards (I think) at some point with an [M], [IS], [T], [R], [D] and a couple of blanks. With the common O and knowing that the M, IS and D were higher-value, I saw MISDOeR and struggled to incorporate the T, finding it difficult to back out of starting the word with a MIS- prefix. It turns out it might have been okay in that I could have found MISsORTeD, there were easier solutions such as aMORTISeD or MODeRnIST.

On hindsight, though, my hinting might not have been ideal. This is because BURrITO and its plural form weren’t the only solutions; there was OBITUaRy, which some people might find easier to see (though I didn’t actually see this during the game). I think clue 1 would probably already be invalid; I knew the word ‘obituary’ in primary school, since it was a section in the newspapers that I would browse through in the mornings, but I’m not sure it would be fair to expect middle school students in general to be familiar with the term. Clue 2 would certainly be incorrect.