Earlier today I asked you to fill in the grid below with entries of the form

[NUMBER][SPACE][LETTER](S)

Such as “Thirteen As”, “Ten Bs”, “Two Cs”, “One D”, with the rule that the entries accurately describe how many letters are in the completed grid. The twelve entries represent all of the twelve letters used in the grid.

I have added numbers to the crossword to make the explanation clearer.

Facebook Twitter Pinterest The grid. With numbers.

To prep this puzzle it is worth listing number words by word length:

Three letters: one, two, six, ten



Four letters: four, five, nine



Five letters: three, seven, eight

Six letters: eleven, twelve, twenty

Seven letters: fifteen, sixteen

Eight letters: thirteen, fourteen, eighteen, nineteen

STEP 1

The shortest entry ( 8 Down) has five units and so it must be of the form “ONE *” since all numbers greater than one will include a plural and require at least six units.

10 Across is six units, so it must include a three-letter number greater than one, and whose final digit is *. The options are one, two, six and ten. Yet one, two and ten would make 8 Down either “ONE E”, “ONE O” or “ONE N” which are all contradictions! By elimination, 10 Across is “SIX #s”

STEP 2

4 Down has an 8-letter number word, so it must be either THIRTEEN, FOURTEEN, EIGHTEEN or NINETEEN Ss.

We can eliminate FOURTEEN and NINETEEN because there are no five letter number words beginning in F or N that we would need for 4 Across.

We know that there are a total of twelve entries, and that only one is singular (8 Down). The final S in each of those entries accounts for 11 Ss on the grid. There is an extra S for the SIX, bringing us up to 12 Ss. The only chance for more Ss would be if there are more entries with a SIX, a SEVEN, a SIXTEEN or a SEVENTEEN. But since there are no free entries for 3-letter, 7-letter, or 9-letter words, we can eliminate SIX, SIXTEEN and SEVENTEEN, so the extra Ss must come from SEVEN. There are only three possible entries where SEVEN fits, so there are at most 15 Ss. So we can eliminate EIGHTEEN as a candidate for 4 Down, and consequently the entry must be THIRTEEN Ss.

Which means 4 Across must be THREE ?s, and 1 Down must be FOUR.

STEP 3

If there are thirteen S’s, with twelve already accounted four, then from above there is a single entry with SEVEN. The only slots that fit are 6 Down and 7 Across. If it was in 7 Across, this would make 3 Down a number of E’s. The options are FOUR, FIVE or NINE Es. There are already seven E’s on the grid (including the SEVEN), so we can eliminate FOUR and FIVE. And we can eliminate NINE since this would mean 4 Across is THREE N’s, which is a contradiction since there would be four Ns on the grid. So, 6 Down is SEVEN !s

7 Across is either THREE or EIGHT.

We have now used 12 letters: EFHINORSTUVX. Since we know there are only 12 letters in the grid, we can eliminate EIGHT since that has a G. So, 7Across is THREE Vs.

STEP 4

3 Down has to be FOUR Hs because if it was FIVE then 4 Across would be repeating the number of Vs.

9 Across must be THIRTEEN Es because Es are the most common of the remaining letters, and we can eliminate FOURTEEN (since this would make 5 Down a number of Us, and Us are already counted in 4 Across) and EIGHTEEN (since there are no Gs) and NINETEEN, since there cannot be as many as nineteen Es in the remaining spaces.

The three remaining numbers each have four letters. Since there are THREE Vs but only one V on the grid, two of the remaining entries must be FIVE. And since there are THREE Us, and two Us on the grid, the final remaining entry must be a FOUR.

So, there are four Os in total, so 2 Across must be FOUR Os, 5 Down must be FIVE Is, 3 Across FIVE Fs and 10 Across SIX Ts. And the remaining spaces in 1 Down and 6 Down must be N and R.

Well done to those who solved it. Gold stars especially to those who tweeted me the correct answer, with (incomprehensible) workings!

Ellen Fay (@EllenFay7) @alexbellos my working out is incomprehensible but I think this is right? (A few lucky 'guesses' going on) pic.twitter.com/R2cgtfGiRu

This puzzle initially appeared in the American Math Monthly in 1987. Thanks to Lee Sallows for his permission to use it.

I post a puzzle here on a Monday every two weeks.

My most recent book is the colouring book Snowflake Seashell Star, which is out this week in the US with the grander title, Patterns of the Universe.

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And if know of any great puzzles that you would like me to set here, get in touch.



