Finding the Number of Cubes in the Middle of a Decreasing Cube Tower

[chewtoy929]

if you where to take something like a sugar cube stack it into a tower 6 cubes high than on each of the 4 sides stack sugar cubes in a decreasing amount so there are 6 cubes in the middle and 5 surrounding it on the four sides , then 4 surrounding that then 3, and so on. What I need is an equation for how to find a tower that has "x" amount of cubes in the middle. Any help?

 

[HallsofIvy]

So you have x cubes in the center stack, x-1 in the stack immediately to the right, then x-2, ..., on down to 1. Not counting that first stack, we have four stacks of x-1, x-2, ... cubes so that part is $4\sum_{i=1}^{n-1} i$. Counting the center stack, that is a total of $x+ 4\sum{i=1}^{n-1}$. There is a standard way to find that sum: it is 1+ 2+ 3+ 4+ ...+ (n-2)+ (n-1). Reverse that sum and you have (n-1)+ (n-2)+ ...+ 4+ 3+ 2+ 1. Add those two "term by term": 1+ (n-1)= n, 2+ (n-2)= n, 3+ (n-3)= n, ..., (n-2)+ 2= n, (n-1)+ 1= n. That, is each of those sums is n and there are n-1 sums. Notice that you have added the sum twice. You have to take that into account.

 

[chewtoy929]

explain like I am 10

 

[matt grime]

Are you 10? (This is important: if you are then explaining it in such abstract terms might be inappropriate.)

You gave the answer there at most 3 minutes thought. It may take you a little longer to digest than that. So try looking at it and thinking about it, perhaps by doing some smaller examples like 1 in the central tower, then 2, then 3...

 

[chewtoy929]

no I am not it was an expression, but I don't understand the answer to the question in the first place, I think I can figure it out though, thanks.