The Critical Line: Volume 9

Annie the actuary is heading to the company Christmas party, and she’s up for a big night. Can you solve the cryptic clues to work out which 12 drinks she’ll be having, and then fit them all into her glass?

Whole words fit in no particular order into the areas with solid lines. Clues for the letters in each row can be found to the right of the glass. There’s more than one way to fill up Annie’s glass, so grab a drink of your own and get creative!

1. Start without existing (3)
2. Go back on half-muttering (3)
3. Kale doesn’t have much potassium (3)
4. Sounds like a cheer for the first bee! (4)
5. Place first exponentially (4)
6. Not stirred, without the ends of heroin (4)
7. Not starboard (4)
8. Scrambled woman (4)
9. By any other name, acutely! (4)
10. Jumbled book (4)
11. Commotion in the animal enclosure with some uranium (4)
12. Reverse into the canyon without any Euros (4)

The Critical Line volume 8 solution

By Dan Mayoh (dan@fintega.com)

The challenge in Critical Line Volume 8 was inspired by the art of Piet Mondrian, a Dutch painter famous for painting coloured rectangles on white backgrounds. However I didn’t want to give away this information when setting the challenge, as a quick google of “Mondrian art puzzle” would have revealed all of the answers! This challenge was not an original creation of mine – I first came across it thanks to the work of mathematician Gordon Hamilton and his Numberphile YouTube channel.  Again though, I couldn’t give credit for that in the original challenge without making the solution easily searchable.

The challenge of finding the minimum score for a n*n square is essentially solved via exhaustive search, however shortcuts and clever thinking can be employed that make the optimal solution able to found by pen and paper rather than needing a computer.  Taking the 6*6 case, start by making a list of all the possible rectangular areas we can use.  These are 1*1=1, 2*1=2, 2*2=4, 3*1=3, 3*2=6, 3*3=9, 4*1=4, 4*2=8, etc.  There will be 15 in all (and $$n$$ * $$\frac{(n-1)}{2}$$ generally), each representing a rectangle of distinct dimensions, with areas ranging from 1 to 36 (1 to n² generally).

Next, we want to find a subset of these 15 areas that add up to 36, and then finally we want to verify if that is a valid subset by seeing if we can arrange the underlying rectangles into a 6*6 square.  If so, we have an answer.  Computer searching can greatly speed things up though, and finding the minimum score for an n*n square makes for a good programming challenge.

Now what is really interesting is that in the few weeks between the time this challenge was published in the Critical Line and the time that this solution discussion was published, new lower results have been found for several values of n!  The on-line encyclopedia of integer sequences lists the lowest scores for n$$\geq$$3 as sequence A276523 and the links and extensions given there show several improved results from the last week of November 2016.

The lowest answer for the 17*17 case is 8, and a configuration of this score is given.

The winner of the book voucher is Marcus Burton, who provided an arrangement giving a score of 14 using three rectangles (6*17, 11*8 and 11*9).  Congratulations Marcus for giving the challenge a go!

– Dan Mayoh

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