by Andrew Skidmore
Published Sunday September 18 2022
Liam has nine identical dice. Each die has the usual numbers of spots from 1 to 6 on the faces, with the numbers of spots on opposite faces adding to 7. He sits at a table and places the dice in a 3×3 square block arrangement.
As I walk round the table I see that (converting numbers of spots to digits) each vertical face forms a different three-figure square number without a repeating digit.
As Liam looks down he sees six three-digit numbers (reading left to right and top to bottom) formed by the top face of the block, three of which are squares. The total of the six numbers is less than 2000.
What is that total?
Sunday Times Teasers Discussion Group 
It appears that Liam is using left-handed dice if he can see three squares on the top face of the block.
Yes, that is a bit naughty!
I don’t think it’s naughty really, the only thing that’s required is all the dice used are identical.
When I started on this teaser, I realised there were two possible ways to arrange the pips on the dice, but did not know these were called Right & Left handed. Wikipedia defines these, and tells me that most Western dice are right handed but Chinese dice are normally left handed.
Don’t understand the question – if a vertical face shows the following spot count
3,4,6,1,2,5,6,6,4
is 316 and 426 and 654 the three digit numbers ?
and how do I get a single 3 digit number from those 9 numbers, please ?
Hi Robert,
Each of the four vertical faces (front, left, back and right) shows only three of the dice. Directly facing each face in turn and reading the number of spots from left to right gives a three digit number that has to be a square with no repeated digits. So there is just one square on each vertical face.
On the top you see three rows and three columns. The three rows give your three three digit numbers reading left to right and the three columns give you three more reading top to bottom.. Three of these six numbers are squares (not necessarily unique and not necessarily having different digits).
Ah – I “read” 27 dice I am now 77 and perhaps it is time to retire…..
Some of us are over 80, but I still managed to get the same result as Brian. It’s important to keep those little grey cells working . . .
I didn’t think the wording was very clear. ‘He places the dice in a 3×3 square block arrangement’ doesn’t make it clear that the square block is flat on the table, just one die thick. So like Robert Farey, I was originally confused about the ‘vertical faces’. I eventually realised that these ‘vertical faces’ were each 3 dice wide and 1 die high. It’s quite easy to find the four different 3 digit squares, but of course we have to find what order they are in. That defines each of the 4 corner dice, and before seeing Cruciverbalist’s comment, I realised that these could follow one of two conventions, although of course we do know that they are all the same. So we have several options to try, and need to find the one arrangement that allows 3 squares within the 6 rows and columns when viewed from above.
I had no problem with the side faces.
Like Cruciverbalist I found that the dice must be “left handed” to permit Liam to get three squares on the upper face of the block. The upper faces of the corner dice are then defined as also noted by Robert Brown.
But Liam can look from four directions. This made the puzzle for me a bit more laborious. But eventfully I found a solution.
Yes. Only one of these four sums to < 2000.
I looked at the top face of the block from four directions and tried to enter the the smallest possible numbers for squares and remaining numbers.
I had a table of squares and payed particular attention to 1st and 3rd digits.
One view did not permit three squares. My smallest totals for the three other views were the solution (below 2000), 2002 and 2101..
There are eight 3X3 top face arrangements which satisfy the constraints but only one summing to <2000. Also, at least 4 dice are of the same “handedness”.
Hi xgha,
“Liam has nine identical dice”, so no worry about a mixture of R- and L-handed.
Hi Peter, Yes, I realised that but was just pointing out that without that information one could deduce that at least 4 dice were identical.
For 9 dice of 2 possible types one can deduce that at least 5 are identical.
Sorry, I meant at least 4 of the “correct” type.
Ah, much easier than last week’s teaser. But I did enjoy it. It was one where I imagined there would be many more possibilities to filter through than there actually were. Vertical faces are easily determined. Then there are 2 ways to fix the corners of the 3×3 square and 4 ways to view each. Could only get 3 square numbers into 4 of these 8 arrangements, but in some cases more than one possible set of 3 square numbers. I had 11 possibilities but only one had a total of the 6 numbers under 2000.
Hi Nicky, I think there are (at least) 8 arrangements and these all use the same set of corner digits.
They all have the same corner digits for sure. But counting them can depend what is meant by an ‘arrangement’. I have 11 if I define it as including 3 squares in the 6 numbers and it mattering which of the 4 orientations you look from. However if I count the different totals of the 6 numbers I only have 10 as two of the arrangements produce the same 6 numbers and total. Looking from the four orientations my 11 break down into sets of 1,4,4,2, the answer being one of the 2. If I then just take away the orientation used to find the six 3 digit numbers to sum, I have just 5 physically different arrangements of the 9 dice.
Nicky
I think there are 12 arrangements, with the 4 orientations giving 2, 4, 4 ,2 arrangements (including 2 with top corners 4,1 ).
For a complete solution showing uniqueness of the answer all 12 must be considered.
Two arrangement have 3 different squares.
Yes, I think it might have been neater to ask for one of the two where the three top squares are all different.
Yes you are right, I had missed one, there are 12.
How perceptive that you seemed to know I had missed one of the two with top corners 4 and 1.
Not really very perceptive. You had said that your 2 was the answer and one other.
By “arrangement”, I mean a view of a 3X3 array which satisfies all conditions. My 8 arrangements fall into 2 sets, with 4 arrangements per set. A set can be defined as a sequence of 8 digits (using 1-6 only) which run clockwise around the 8 edge dice. The 4 arrangements for a particular set are then found by starting the sequence from each of the 4 corners of a blank 3X3 array. I have two distinct sets which work. Writing them as starting in top left corner they are similar but have second and sixth digits swopped over. All corner dice in each of the 8 arrangement come from the same set of 4 digits and the the centre dice face is always 4. The four corner dice have the same specific handedness and the other five could be random – but we are told all dice are the same handedness. To find a unique solution this information was not really needed.
Ah, I have 12 after Tony encouraged me to find another 1.
I have your 8 but there are another four. They have a 2 in the centre and involve three more sequences of 8 numbers as you describe around the centre.
Yes, thanks Nicky and Tony. I have now found all 12 arrangements with the corresponding 5 edge sequences providing 4, 4, 2, 1 and 1 of these. Going back through my scribbles now realise what happened. Having established centre dice face had to be 2 or 4 (5 and 6 quickly ruled out as options) I started with 2. Unfortunately, the first arrangement I found with 3 squares was the sequence 56441553 starting in top left corner but then saw adjacent 5 and 2 dice in bottom mid-edge was not allowed. Moving to centre dice 4, I quickly found a/the solution (<2000 sum) and didn’t go back to check for uniqueness. Careless!
xgha
The 4 arrangements which you seem not to have found have a central digit which is not 4.
All 12 arrangements need to be considered to prove the answer is unique.
Does anyone know where to purchase left-handed dice in the style of the right-handed dice that we use here in the UK?