Identical blocks are stacked in rows to create a tower 24 rows tall. If the top row of the tower consists of four blocks, and each row below the top row consists of eight more blocks than the row directly above it, how many blocks are in the entire tower?
By my calculation, the total number of blocks is 2304. We know the tower is 24 rows tall. We know that the top row of blocks is 4 across, which means that there are 23 rows that are also 4 across directly underneath the top row. I've shown this in the above graphic by shading that area gray.
If the top row can be labeled as Row 1, then the row directly beneath will be Row 2 and so on. Row 2, then, consists of 4 blocks + another 8 blocks. Row 3 would then be 4 + 16; Row 4 would be 4 + 24, etc.-- and this goes on for 23 rows.
How to deal with this mess?
As you see above, I took Row 1, and the first 4 blocks of every row after that, to be a single column, i.e., 4 x 24 blocks, which equals 96. This leaves me only the light green area to deal with. If we redefine Row 2 as having 8 blocks, Row 3 as having 16 blocks, etc., we see this progression:
(Row 2) 8 x 1
(Row 3) 8 x 2
(Row 4) 8 x 3
...
(Row 24) 8 x 23
We're adding all these rows together to get the total number of blocks in the green area, so the distributive property is useful here. We factor out the 8, and we see that the number of blocks in the green area will be
8(1 + 2 + 3 + ... + 23)
So how the hell do you do the sum of 1 through 23 quickly? You can add everything up old-school-style, or you can use Karl Gauss's method:
For any sum (1 + 2 + ... + N), the quantity is (N + 1)(N/2).
That means, since we're going from 1 to 23, that
(23 + 1)(23/2)
is what we're looking for. That's equivalent to 23*12, which is 276. But we can't forget to multiply by 8 (distributive property!), so
276*8 = 2208.
So that's the number of blocks in the green area. Add that to the 96 blocks in the gray area, and we get 2304.
QED.
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