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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