# Sunday Times Teaser 2743 – Line-Up

*by Nick MacKinnon*

#### Published: 19 April 2015 (link)

I have arranged the numbers from 1 to 27 in a 3-by-3-by-3 cubical array. I have noticed that the nine numbers making up one of the faces of the cube are all primes. Also, I have searched through the array and written down the sum of any three numbers that are in a straight line. I have then calculated the grand total of all those line-sums. It turns out that the grand total is itself a perfect cube!

What is that grand total?

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This is an update of my previous attempt, which was far too slow. In this version, rather than searching for an exact solution directly, I find a ‘near miss’ and then use this as a base for swapping pairs of numbers until an exact solution is found.

Here is an example solution:

IMHO to have solved this teaser fully, one has to demonstrate that there actually is a way of arranging the 27 numbers as specified.

Without that, one finds the answer assuming that there is a solution.

Needless to say (I’m not that humble after all) I have done so.

Hi Peter,

I agree and my program above does do this although I don’t output the sequences that it finds.

We can find upper and lower limits to the possible total sums, the lower being 1734 and the upper is 2550, the only cube in this range is 13^3. I suspect there are a few arrangements that fulfil the criteria too.

Just to show I’m not all talk:

TOP//MIDDLE//BOTTOM as follows:

13 02 17 // 01 15 04 // 22 10 24

03 07 05 // 16 09 18 // 12 21 14

19 11 23 // 06 20 08 // 25 27 26