Word Grids

I try to design activities every once in a while that have a tie-in to another discipline. Here we'll look at a collection of puzzles that are superficially about words but are more about geometry and arithmetic once you start looking for strategies.

Setup

In order to run this as an activity, you'll need a collection of word sets that can fit into grids. I've provided some samples below and some more here. Small crosswords can also be a good source, but sometimes you have to hunt for a good one or stretch your definition of "word." In each puzzle, the task is to place words into the grid so their letters fill all boxes in a single row or column reading left-to-right or top-to-bottom.

Example 1

Place all 8 words into the 4x4 grid:

AGES HIGH IDEA PEST SHIP SWAT THAT WIDE

Example 2

Place 8 of the 9 words into the 4x4 grid:

ACHE EACH ECHO SEAT STET TACO TEST THEN TOON

Example 3

Place 8 of the 10 words into the 4x4 grid:

ACAI ACED CARE CART EDDY RAID REIN PAIN PITY TINY

Example 4

Place all 16 words into the two 4x4 grids:

ACNE AURA CLAY EELS EYES MANY ORAL ORCA

PACE PROW SOME RAYS RULE STAR TRAY WAYS

Initial Questions

• What are some strategies for placing the first word in the grid?
• If there are words you will place in the grid and words you will leave out, how can you decide which are which?
• If the words must be divided amongst two or more grids, is there a way to tell which words should go in which grid?
  

Explorations

Here are some observations students might make:

• There is symmetry about the main (downward) diagonal. If you can make a solution with a certain word as the kth row, you can make a solution with that word as the kth column by mirroring across the diagonal and vice versa.
• If there is a letter that shows up in exactly two words, then that helps you enter those words pretty easily.
• For any choice of i and j, if there is only one pair of words where the ith letter of the first matches the jth letter of the second, then they must be put in the jth row and ith column (or, by symmetry, jth row and ith column), respectively. Students will likely see this for i = j = 1 first -- i.e., if there are only two words that start with the same letter, they must make up the first row and column.
• Since every position in the grid is shared by two words, every letter in the grid is doubly-represented in the word list. In particular, if you count up the number of a given letter amongst the words in your grid, you must get an even number.

One way to organize large word sets is to create a graph. For example, if you write down every word in the set and draw an edge between pairs of words where you know inclusion of one requires inclusion of the other, then you can try to form your grids from the connected components. A directed graph could similarly be useful for tracking cases where inclusion of one implies inclusion of the other, but not the converse.

Wrap-Up Questions

• What tricks did you find to solve these puzzles? Is there anything you now look for first?
• What makes one puzzle easier or harder than another? Are there any features you can point to or do you just get a sense for it while solving?
• Is there an algorithm for solving these puzzles? (i.e., is there a set of steps to follow that always solves them?)

Extensions

Students often like to explore making their own puzzles in activities like these. While this is pretty tricky to find sets of real words that can be put into a grid, especially for longer words. Four letters is probably around most students' limit, but they could make some nice three-letter-word 3x3 grid puzzles.

You can still get a lot out of playing with "words" that are just strings of letters. For example, with the word set ABCD, AEIM, BFJN, CGKO, DHLP, EFGH, IJKL, MNOP, it's pretty straightforward to get the solution

A	B	C	D
E	F	G	H
I	J	K	L
M	N	O	P

or its mirror because each letter appears exactly two times: once apiece in two different words. On the other hand, a word set like AAAB, AABB, ABBA, ABBB, BAAB, BABB, BBAB, BBBB is quite a bit tougher to work with and has a unique solution, up to reflection. Using only a handful of letters for your words, you can already design some pretty challenging puzzles. The word set AAAC, ABBC, ACED, BAAD, BEAE, CAED, CDEB, CEAC, DADD, DDEB, EAAE, EABD can make one 4x4 grid with four words left over, for example.

Trying to make a puzzle that has exactly one solution but where that solution is difficult to find is tough!

This post was sponsored by the Julia Robinson Mathematics Festival