[bonanova]

A grocer had some oranges he wished to display.

Inspired by a recent Brainden puzzle, he layered his stack of fruit using triangular numbers. On top was a single orange. Beneath that was a layer of three oranges. In turn, the next layers had 6, 10, 15, 21, 28 and so on. until the oranges were used up. The entire stack, fortuitously, comprised a tetrahedron - a perfect triangular pyramid.

Until it was struck by the careless shopper making an illegal cell phone call while operating her shopping cart.

Faced with the task of reconstructing the citrus tower, the grocer opted for what he hoped was an easier task - to make two pyramids, unequal and smaller, but both still making perfect triangular structures.

How many oranges comprised the two pyramids?

Hint: there couldn't have been fewer.

[plainglazed]

hey bonanova - question, is the unequal bit a requirement?

[Guest]

680

[Guest]

Since they can't be the same size it can't be taking a stack of 20 and turning it into 2 stacks of 10.

The next number where the bottom level of oranges equals the exact number of oranges in a smaller pyramid is...

The original stack was 680. If you take the bottom layer off that stack (120 oranges) it will create a pyramid 8 oranges tall, and reducing the original pyramid from 15 tall to 14 tall (560 oranges)

[Guest]

nevermind

Edited November 2, 2010 by enrightmcc

[Guest]

680

While technically you are correct in the number of oranges that comprised the two pyramids in total, I think the original question was asking how many oranges in EACH pyramid.

[Guest]

Since they can't be the same size it can't be taking a stack of 20 and turning it into 2 stacks of 10.

The next number where the bottom level of oranges equals the exact number of oranges in a smaller pyramid is...

The original stack was 680. If you take the bottom layer off that stack (120 oranges) it will create a pyramid 8 oranges tall, and reducing the original pyramid from 15 tall to 14 tall (560 oranges)

Did you solve it by hit and trial or analytically?

Analytically, I arrive at the below equation and cant seem to find a solution or a simplification from this.

Crosschecking them for correctness, these equations do hold valid for your solution of 680 (560 + 120) and the other solution with 20 oranges with 2 heaps of 10 each.

n(n+1)(n+2) = a(a+1)(a+2) + b(b+1)(b+2)

where n is the height of the original tower and a and b are the heights of the 2 new towers.

[Guest]

Since they can't be the same size it can't be taking a stack of 20 and turning it into 2 stacks of 10.

The next number where the bottom level of oranges equals the exact number of oranges in a smaller pyramid is...

The original stack was 680. If you take the bottom layer off that stack (120 oranges) it will create a pyramid 8 oranges tall, and reducing the original pyramid from 15 tall to 14 tall (560 oranges)

Did you solve it by hit and trial or analytically?

Analytically, I arrive at the below equation and cant seem to find a solution or a simplification from this.

Crosschecking them for correctness, these equations do hold valid for your solution of 680 (560 + 120) and the other solution with 20 oranges with 2 heaps of 10 each.

n(n+1)(n+2) = a(a+1)(a+2) + b(b+1)(b+2)

where n is the height of the original tower and a and b are the heights (number of layers) of the 2 new towers.

[Guest]

deegee, the total number of oranges increases like so...

1,4,10,20,35,56...

the formula should be...

a +b +c +d = 1

8a +4b +2c +d = 4

27a +9b +3c +d = 10

64a +16b +4c +d = 20

37a +7b +c = 10

19a +5b +c = 6

7a +3b +c = 3

18a +2b = 4

12a +2b = 3

6a = 1

a = 1/6

2b = 1

b = 1/2

7/6 +3/2 +c = 3

16/6 +c = 3

c = 1/3

1/6 +1/3 +1/2 +d = 1

d = 0

so...

1/6 *n^3 +1/2 *n^2 +1/3 *n = total oranges

where n is number of layers.

[Guest]

analytically, we have..

where 1/6 *n^3 +1/2 *n^2 +1/3 *n and 1/2*n*(n+1) evaluate to same value.

[Guest]

Pretty sure this is it.

[Guest]

nevermind

Edited November 2, 2010 by cifus

[bonanova]

hey bonanova - question, is the unequal bit a requirement?

Ah, you know it would be too simple if they were equal ...

[plainglazed]

Okay well maybe not in the spirit of the OP but thinking a little outside of the crate -

If the original pyramid had 120 oranges, two pyramids can be constructed of 56 and 35 oranges sharing a corner (1 orange).

This is the first instance this occurs where the two pyramids are different sizes and the second pyramid is not wholely

contained within the first pyramid (also a requirement of the OP that each is smaller than the original). That is to say

where the difference between the number of oranges in the original pyramid and the two pyramids is a perfect pyramid.

[bonanova]

:-) gets the number of oranges [which OP ambiguously asks for]

enrightmcc gets the three pyramid sizes

phillip1882 gets the analysis