Cannonball Puzzle

[rosevallen]

Hello! I was reading a book called the Joy of Mathematics by Theoni Pappas (for those curious) when this cannonball entry came up. I counted the dimensions and found a way to express it, but I’m not yet educated enough in math to know if there’s a more elegant way to express it. Does anyone know a simpler way to write this? The only already existing formula I could find was the Diophantine Equation for a square pyramid; this one is rectangular. It’s base is 12x15(w by l), it’s height is 12, and at the top is 1x8. I counted the first two layers to see how the pattern changes from the bottom up; I found that width lost 2 and length lost one so I came up with this -

• (12x15)+(10x14)+…(1x8) = 568.

In summary, I’m wondering;
A) is there a better way to write things like this once you know the pattern is w-2 l-1 repeating 6 times, a way of getting rid of those dots?
B) does an equation for rectangular pyramids in this case already exist/has google failed me?

Thanks for any help anyone is able and willing to provide. I’m just curious

 

[lookagain]

It’s base is 12x15(w by l), it’s height is 12, and at the top is 1x8. I counted the first two layers to see how the pattern changes from the bottom up; I found that width lost 2 and length lost one so I came up with this -

• (12x15)+(10x14)+…(1x8) = 568.

Why is the top 1 by 8? That does not fit the pattern of reducing the previous width by 2, and it also does not
reduce the previous length by 1.


(12*15) + (10*14) + ... + (1*8) = 568 <---- This is
a correct sum, but the (1*8) is inconsistent.

 

[Dr.Peterson]

Hello! I was reading a book called the Joy of Mathematics by Theoni Pappas (for those curious) when this cannonball entry came up. I counted the dimensions and found a way to express it, but I’m not yet educated enough in math to know if there’s a more elegant way to express it. Does anyone know a simpler way to write this? The only already existing formula I could find was the Diophantine Equation for a square pyramid; this one is rectangular. It’s base is 12x15(w by l), it’s height is 12, and at the top is 1x8. I counted the first two layers to see how the pattern changes from the bottom up; I found that width lost 2 and length lost one so I came up with this -

• (12x15)+(10x14)+…(1x8) = 568.

In summary, I’m wondering;
A) is there a better way to write things like this once you know the pattern is w-2 l-1 repeating 6 times, a way of getting rid of those dots?
B) does an equation for rectangular pyramids in this case already exist/has google failed me?

Thanks for any help anyone is able and willing to provide. I’m just curious

The reference to a Diophantine equation is really irrelevant; they are just using the formula for a pyramidal number to form a Diophantine equation to solve "the cannonball problem", which is not the same as your problem.

If you had been given that formula in the problem you are working on, you could use it as part of your work, by viewing the figure as a pyramid attached to a triangular prism. I'm not sure how they expect you to use what they show, which doesn't include such formulas.

As far as I can tell, they just want you to make a summation. But it seems to me that both dimensions of each layer are 1 less than the layer below. Check that out.