### Brain Teasers

# Mountain of Love

Once upon a time, in the Kingdom of Snow, there lived the fair Princess Miyuki, who had always been fascinated by fairy tales and the stories of the wonders of the world. She believed that her sleeping, handsome prince was cursed to be one of the mummies in the pyramids of Egypt, and when she was done being Fantaghiro-saving-her-Romualdo, her prince would then build a beautiful palace as a monument of their love (which of course would be more beautiful than Taj Mahal) where both of them will live happily ever after...

So, you see, that's why Princess Miyuki had refused each and every love declaration or marriage proposal that came her way.

But of course, what's a story without the brave and wise heroic prince? Here comes Prince Hajikidama of the Marbles Kingdom. He's no mummy, but his brilliant(?) brain deduced that in order to win the princess' heart, he'd have to give her what she wanted most - her fantasy, and his token of understanding (her childish delusional nature).

In the end, labouring his mind's creativity department for all its worth, Prince Hajikidama decided that a 'Mountain of Love', reaching so high it collected the never-fallen-snow in Marbles Kingdom, might do the trick. That actually is a tetrahedral-shaped (pyramid with 4 equilateral triangular faces) crystallic marbles-made structure (like a giant pile of apples or oranges at a greengrocer's store) consisting of one marble crowning on the top layer, three on the next-to-top layer, six on the third layer, ten on the fourth layer, and so forth. If there are exactly 1,000,000 layers, specify the total number of marbles needed in the construction of entire mountain.

So, you see, that's why Princess Miyuki had refused each and every love declaration or marriage proposal that came her way.

But of course, what's a story without the brave and wise heroic prince? Here comes Prince Hajikidama of the Marbles Kingdom. He's no mummy, but his brilliant(?) brain deduced that in order to win the princess' heart, he'd have to give her what she wanted most - her fantasy, and his token of understanding (her childish delusional nature).

In the end, labouring his mind's creativity department for all its worth, Prince Hajikidama decided that a 'Mountain of Love', reaching so high it collected the never-fallen-snow in Marbles Kingdom, might do the trick. That actually is a tetrahedral-shaped (pyramid with 4 equilateral triangular faces) crystallic marbles-made structure (like a giant pile of apples or oranges at a greengrocer's store) consisting of one marble crowning on the top layer, three on the next-to-top layer, six on the third layer, ten on the fourth layer, and so forth. If there are exactly 1,000,000 layers, specify the total number of marbles needed in the construction of entire mountain.

### Answer

The formula isS = n(n+1)(n+2)/6

so when n = 1,000,000

S = 166,667,166,667,000,000

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## Comments

Yes, that is perfectly correct.

To explain it better, the triangle numbers 1,3,6,10,15,... have the n'th term .5n^2+.5n

So we need to sum this from 1 to 1000000

the sum of n^2=(1/6)(n)(n+1)(2n+1) [from 1 to n]

and the sum of n=(1/2)n(n+1) [from 1 to n]

Putting it together gives the formula stated in the answer

To explain it better, the triangle numbers 1,3,6,10,15,... have the n'th term .5n^2+.5n

So we need to sum this from 1 to 1000000

the sum of n^2=(1/6)(n)(n+1)(2n+1) [from 1 to n]

and the sum of n=(1/2)n(n+1) [from 1 to n]

Putting it together gives the formula stated in the answer

Anyone care to demonstrate how these formulas are obtained? I guess we just have to take your word for it that they are correct eh?

Yep, you'll just have to take my word for it, because I got better things to do than explain every single little calculation.

I gave up on trying to figure out the formula. I did know that it would be equal to the sum of the even squares. So I just used an Excel spreadsheet to square all of the even numbers from 2 through 1000000, then summed those results. The answer is correct.

forget attempting the equation . i was too stinking lazy to even read it .

me, too -- brain teasers, not essays!

I guess I didn't read it carefully enough. When I read "pyramid with 4 equilateral triangular faces", my mind said "square pyramid. (Also, since this is the shape of all of the ancient pyramids, it was already in my mind.)

The formula for square pyramid numbers is (2n^3 + 3n^2 + n)/6, so THAT answer is 333,333,833,333,500,000.

The formula for square pyramid numbers is (2n^3 + 3n^2 + n)/6, so THAT answer is 333,333,833,333,500,000.

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