Six Numbered Cubes

We had two lengthy files of explanation which are well worth looking at from Tom and Andrew who attend the British School of Paris.  See them here: Andrew.doc or  Andrew.pdf  and  Tom.doc  or  Tom.pdf.

Mali, who goes to St. Philip's School in Cambridge England, said:

The highest total is 83. The largest numbers 4, 5 and 6 must go on the very top to make the total as large as possible.
The lowest total is 64. Now, the largest numbers must go at the very bottom to have fewer faces visible.
To get 75 the ladder should be made of 4 on the top, 1 underneath, 2 at the bottom. Next column, the 3 at the top, the 5 at the bottom. Finally 6 on its own.
To find this solution, I cut pieces of paper with the number of visible sides on (5, 3, 3, 4, 4, 2), and pieces of paper with the numbers 1, 2, 3, 4, 5, and 6. I created pairs, multiplied them and add the total. I kept changing the pairs around until I found 75.

We had a lot of solutions sent in from Our Lady of Lourdes School: Stanley, Sean and Hedley; Ailbe; Chelsea; Gabriel, Fintan and Charlie; Catsaneman; Mia and Zoe; Mathias and Ethan, Comjen; Nansai; Ohene and Noeleen, Eene and Joe. They made paper cubes with the numbers on and experimented with different parts of the challenge - well done and thank you for your submissions.

Nathan from Cornelius Vermuyden in Essex, England sent in the following:

These cubes are then stood in a vertical tower in any random order, like so....
A
B
C
D
E
F

Each letter can represent any cube as long as it is there only once. With a tower like this the cubes all have only four of their six faces showing except the top cube, which has five of its six faces showing. This means that we have every cube with four faces and an extra one at the very top of the tower.

So we can see that any six consecutive positive integers (not including 0) when multiplied by 4 (for the number of sides seen) and added together, equal above 80, the lowest being 84 with our current numbers 1-6. If we say A = 1, then adding the final face at the top of the tower will only increase the number made by 1 giving us our lowest possible total of 85. Therefore a total of 80 is impossible. 

Algebraically this can be shown by representing the first of the consecutive numbers with n. Let's put this one at the top of the tower and have the other numbers go consecutively down the vertical tower.

n
n+1
n+2
n+3
n+4
n+5

We then need to allow for the faces on each of our cubes that are shown so we can write our expressions as”¦
5n
4(n+1)
4(n+2)
4(n+3)
4(n+4)
4(n+5)

We then expand the brackets and simplify this to”¦
5n
4n+4
4n+8
4n+12
4n+16
4n+20

Now we can collect like terms to give us 25n+60
Using 1 as our lowest positive integer, substitute n = 1 into the expression 25n + 60 to give 25 x 1 + 60 = 85
Once again, this shows that the lowest possible total is 85.

Sabine, Amelia, Sophie, Liam, Tom, and James, from Hardwick Middle School, sent in their ideas. Here is just a sample from the first three:

For question 1 we found that 78 was our highest total. We made this total by putting 6 at the top, 3 below it and 2 below 3. Next to 3 we put 5 and below that we put 1. Next to 1 we put 4.

For question 2 we got 64. We got this by putting 1 at the top, 5 below 1 and 4 below 5. Next to 5 we put 3 with six below. Next to 6 we put 2.

On question 1 we had to make six be shown the most, then five, then 4. we had to make 1 shown the least, then 2, then three.

For question 2 we had to make 1 show the most and 6 the least.

We had to do this otherwise the answer wouldn't be as low or as high as it could be.

For question 3 we had to get exactly 75 put of a staircase of cubes. It took us a while but we finally got the answer. At the top we put 2, below that was 6 and below that was 3. Next to 6 was five with 1 below it. Next to 1 was 4. That made 75.


This next solution was sent via the Wild.maths site which also has this challenge. It was from Ana G. in year 6 at Oaks Primary Academy and can be viewed as the pdf  here.pdf . 
Thank you all for your work, there was too much to show everyone's workings. I gather that it was generally enjoyed.