This is a tricky challenge but we had some good responses.  Not all of them were completely correct, but most contained excellent thinking and so here are a selection.  If you want to know the 'answers', then scroll down to the bottom of this page.

We had these solutions from C of E Primary School, Ledston.  Firstly, Mia and Millie said:

We got 10 and 43 as the cubes directly underneath cube number one by:
• Counting all the cubes on each layer. Which led us to a pattern of 1,4,9,16,25 etc.
• Imagining it clearly in our heads
• Counting each cube in our heads as we went down
• We wrote the number of each layer like this and put a tick next to the ones we knew it would be (the third and fifth layers)
For the cubes directly beneath 4 we got 24 and 67, and directly beneath 8 we got 39 using the same strategies as before.

Then from Edward, Linden and Sam at the same school we had:

To get the first answer we did:
First we worked out how many cubes are on each layer
Layer 1=1 layer 2=4 layer 3=9 layer4=16 layer 5=25 layer 6= 36           
Visualise how many cubes there are so far going from the top to bottom
Layer 1 =1 layer 2=5 layer 3=14 layer 4=30 layer 5= 55 layer 6=91
Then we found out which layers have a centre.
We found the layers 3 and 6 have centres.
We then find out what cube was in the centre
The answers we got were 10 and 43.

To get the answer for the 2nd question we did:
First we worked out how many cubes are on each layer
Layer 1=1 layer 2=4 layer 3=9 layer4=16 layer 5=25 layer 6= 36           
Visualise how many cubes there are so far going from the top to bottom
Layer 1 =1 layer 2=5 layer 3=14 layer 4=30 layer 5= 55 layer 6=91
Then we found out that it had to be on a layer with an even number
These were the answers we got 24 and 75

To get the third answer we did:
First we worked out how many cubes are on each layer
Layer 1=1 layer 2=4 layer 3=9 layer4=16 layer 5=25 layer 6= 36           
Visualise how many cubes there are so far going from the top to bottom
Layer 1 =1 layer 2=5 layer 3=14 layer 4=30 layer 5= 55 layer 6=91
Find a layer with a number that is in the 8 times tables.
We got the layer 4
We got the answer 33

Finally from Adam and Patrick

This is how we completed the nrich pyramid problem step by step.
First we read the instructions until we understood the problem.
Next we counted the number of squares in each layer starting from the North West corner going right and added them together as we went along; the answers were: 1+4=5+9=14+16=30+25=55+36=91.
We then looked at the first question which was: what number cubes are vertically underneath (in a straight line):
Then we attempted to solve the first question.
Soon we realized that the numbers went in a straight line through the middle of the pyramid.
We found out that only layers that worked were odd.
Those were: layer 3 & 5 and the numbers: 10 & 43                     
After we solved that question we moved on to the next which was the same question just a different starting number.
The next question was harder but we solved it as well.
The answer was: 53 and 13.
The next few answers were similar.
At that point we were on a roll.
For the last 4 questions we worked out they were the same as the first three but the last one was simple.
The answer was no.

Here are the solutions -
Under 1 is 10 (followed by 43), under 4 is 24 (followed by 76), under 8 is 39.
Middle of south 2 up is 53 and 2 at a time takes you to 13 and 1.
South west corner stones are 86, 51, 27, 12, 4 and 1.

Re-numbering the other way gives 82 under 91, under 89 it's 71 and under 80 it's 45.
There are no numbers in the same place with the re-numbering.

But again well done for trying this unusual harder challenge that is purely based on visualising.
If anyone has a further try do send in your solutions.