IQ / Programming Quiz (Cannon-Ball Stacks)

[Mark_Wallace]

I just made one with mints, and I had to put one in the middle on the third layer. Mind you, the mints are all different sizes and shapes -- but if they were all the same, it would have drained the fun out of it.

I wanna be a eunuchs developer! Pass me a bread knife!

[tolw]

My shot at the answer - quick and not optimized, I'm sure :)

class Program
{
    const int RANGE = 1000;

    static void Main( string\[\] args )
    {
        int sum, p1, p2;
        for( int a = 0; a < RANGE; a++ )
        {
            p1 = ( a \* ( a + 1 ) \* ( a + 2 ) ) / 6;
            for( int b = a + 1; b < RANGE; b++ )
            {
                p2 = ( b \* ( b + 1 ) \* ( b + 2 ) ) / 6;

                for( int i = b + 1; i < RANGE; i++ )
                {
                    sum = ( i \* ( i + 1 ) \* ( i + 2 ) ) / 6;
                    if( sum == p1 + p2 )
                    {
                        Console.WriteLine( sum.ToString() + "=" + p1.ToString() + "+" + p2.ToString() + "\\n" );
                    }
                }
            }
        }
        Console.ReadKey();
    }
}

SPOILER ALERT!! The answer I got is: 680 = 120 + 560 EDIT: I am aware I could have just stopped looking after the first hit - but I was also curious if there would be a lot of results :)

[Andre Kraak]

The number of balls for each level of the pyramid increases like: 1, 3, 6, 10, 15, 21, 28. This means that the number of balls for each larger pyramid is as follows: 4, 10, 20, 35, 56, 84. So we are looking for a pyramid size that can be created with the sizes of the two pyramids that go before it. 84 is the first number for which this is possible, being combined from the pyramids with 35 and 56 balls.

0100000101101110011001000111001011101001

[tolw]

From the OP:

(assuming he uses every cannon-ball in both pyramids)

[AspDotNetDev]

Points awarded for: correctness, elegance, quickness, and mathematical insight. It seems you found an equation for the number of elements in a pyramid for the Nth pyramid. Nice! Since you already have a better program than mine, I might as well share mine now instead of waiting until tomorrow:

public partial class Form1 : Form
{
public Form1()
{
InitializeComponent();
for (var i = 1; i <= 30; i++)
{
txtOut.AppendText(Environment.NewLine + CalculateNumBalls(i).ToString());
}
}

public int CalculateNumBalls(int index)
{
    if (index <= 1)
    {
        return index;
    }
    else
    {
        return CalculateNumBalls(index - 1) + CalculateLayer(index);
    }
}

public int CalculateLayer(int index)
{
    int total = 0;
    for (var i = 1; i <= index; i++)
    {
        total += i;
    }
    return total;
}

}

That just displays the number of balls in the Nth pyramid for the first thirty pyramids. I just used a calculator to see what the difference was between each number. I saved time because I didn't have to calculate all the way down (given largePyramid - mediumPyramid = smallPyramid, I could just stop when smallPyramid was larger than mediumPyramid). Out of curiosity, did you know that formula already, or did you figure it out?

Thou mewling ill-breeding pignut!

[AspDotNetDev]

Virtual balls. Also, pool may help. :)

Thou mewling ill-breeding pignut!

[Mark_Wallace]

Oh, I know the Maths, but I don't come to the Lounge for work. It's much more fun ignoring thousands of years' learning on triangular numbers, etc, and building pyramids out of mints. I've already done my serious posting of the day. It's a bit much to expect another one.

I wanna be a eunuchs developer! Pass me a bread knife!

[AspDotNetDev]

You keep deleting your messages before I can reply to them, so here is what I was going to say in response to one of them... :)

---

There is no pyramid composed of exactly 91 balls. It goes 84, then 120. You are on the right track though. Let me phrase it another way. There are 3 pyramids: A, B, and C. A is the smallest, B is larger, and C is the largest. The number of balls in C is equal to the sum of the balls in A and B. A does not equal B. This is the question: Find the smallest possible value of C.

Thou mewling ill-breeding pignut!

[AspDotNetDev]

If they are thin mints, I fear you may never succeed, as the rate of consumption will outpace the rate of production.

Thou mewling ill-breeding pignut!

[AspDotNetDev]

Super Lloyd wrote:

level N has N(N+1)/2 balls

Points for realizing this. I too realized this, but couldn't remember the exact function. :thumbsup:

Thou mewling ill-breeding pignut!

[tolw]

To tell you the truth I had to Google 'tetrahedral' to begin with ( my knowledge of English is limited I'm afraid :) ), which led me to: here[^] and later (through the See Also section) to here[^] Things were kind of obvious from there :) Thank you wikipedia :)

[Super Lloyd]

I deleted them because I finally understood the question and saw that someone answered it! :P

A train station is where the train stops. A bus station is where the bus stops. On my desk, I have a work station.... _________________________________________________________ My programs never have bugs, they just develop random features.

[Mark_Wallace]

I'm going out in a bit to get some dolly mixture, and some jelly babies to build the pyramids. I'll nip up and give your OP a five, because it's really inspired me to be productive, today.

I wanna be a eunuchs developer! Pass me a bread knife!

[AspDotNetDev]

Points awarded for sharing your research techniques, for pointing me to this GIF, and for figuring out a solution so fast without even knowing what a "tetrahedral" is. :)

Thou mewling ill-breeding pignut!

[leppie]

Pascal's Triangle[^] Also see: Pascal's Pyramid[^]

IronScheme
((λ (x) `(,x ',x)) '(λ (x) `(,x ',x)))

[AspDotNetDev]

Points awarded for awarding me points (always a winning proposition). :rolleyes:

Thou mewling ill-breeding pignut!

[AspDotNetDev]

How does that relate to the cannon-ball problem? :confused:

Thou mewling ill-breeding pignut!

[leppie]

AspDotNetDev wrote:

How does that relate to the cannon-ball problem?

Exercise for the reader ;p (Can you not see the symmetry?)

IronScheme
((λ (x) `(,x ',x)) '(λ (x) `(,x ',x)))

[Lost User]

The way I see it, each stack must have a number of cannonballs from the set {1, 4, 10, 20, 35...} Now we are looking for a minimum A and B, both out of this set, so that the sum of both is also in the set. From your example we know it works for two stacks of 10 cannonballs, so the sum cannot be greater than 20. Therefore A and B must both be in {1, 4, 10} Now you can simply try all combinations:

A | B | Sum | Sum <= 20 | Sum in set

1 | 1 | 2 | Y | N
1 | 4 | 5 | Y | N
1 |10 | 11 | Y | N
4 | 1 | 5 | Y | N
4 | 4 | 8 | Y | N
4 |10 | 14 | Y | N
10| 1 | 11 | Y | N
10| 4 | 14 | Y | N
10|10 | 20 | Y | Y

So here we have it. Your example with two stacks of 10 cannonballs already is the minimum. Edit: Added pre tags to make the table readable

I'm invincible, I can't be vinced

[BillWoodruff]

Hi AspDotNetDev, In this comment to your good self(ves?)[^], I think you raised an issue appropriate for "Site Suggs and Buggs:" and I have expressed my views there:[^]. And, to this starving worm on a wilted cabbage leaf, the issue raised by that post is an even more compelling puzzle than the stacking of cannon-balls, although I am not denigrating the value of considering the stacking of cannon balls ! best, Bill

"Science is facts; just as houses are made of stones: so, is science made of facts. But, a pile of stones is not a house, and a collection of facts is not, necessarily, science." Henri Poincare