IQ / Programming Quiz (Cannon-Ball Stacks)

[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

[AspDotNetDev]

Nope, there was a stipulation that A and B are different. Otherwise, you are one the right track.

Thou mewling ill-breeding pignut!

[AspDotNetDev]

leppie wrote:

Can you not see the symmetry?

They both involve triangles. Other than that, nope.

Thou mewling ill-breeding pignut!

[Thor Sigurdsson]

A=20, B=54, C=55 or B=54, A=20, C=55 used two recurisve functions and iteration (lazy mans search).

I=I.am()?Code(I):0/0;

[jsc42]

I did this using an old technique: pencil and paper. It is more complicated to explain than it was to do. Basically, I wrote 3 columns of numbers: Col 1: 1, 2, 3, 4, etc [represents the no of levels in the pyramid) Col 2: 1, (corresponding no in col 1 + prev number in col 2), ... [represents the no of new balls in the new level of the pyramid] Col 3: 1, (corresponding no in col 2 + prev number in col 3), ... [represents the total no of balls in the pyramid] and then looked for a number in col 2 that was the same as a number in col 3 which is a pyramid's worth of balls that is also a level's worth of balls [Actually, I didn't bother with col 1 - I could work that one out without writing it down, but it is easier to explain with it there] My solution is ... The first one found was 120, so the solution is 120 + 560 (no in col 3 before the 120 in col 2) = 680 (no in col 3 next to 120 in col 2). (select the text in the gap above, e.g. by dragging the mouse, to read it) Of course, this discounts the possibility that a pyramid might split across multiple levels in a combined pyramid. Just in case I had made a simple arithmetic error, I checked in MS-Excel: Cell A1 = 1, B1 = 1, C1 = 1 Cell A2 = =A1+1, B2 = =A2+B1, C2 = =B2+C1 Cells A3 through C17 = Copy and paste A2 through C2 [I've had to edit this entry twice - suffering from lysdexia (!) today].

[YvesDaoust]

1. looked up the tetrahedral numbers in Wikipedia and found this list: 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969... (10 seconds of work) 2) opened an Excel sheet and created the table of sums 2 5 11 21 36 57... 5 8 14 24 39 60... 11 14 20 30 45 66... 21 24 30 40 55 76... 36 39 45 55 70 91... ... (just needed to recall the appropriate syntax to form the cell expression; 1 minute of work) 3) sorted all sums in increasing order 2, 5, 5, 8, 11, 11, 14, 14, 20, 21, 21, 24, 24, 30, 30, 36, 36, 39, 39, 40, 45, 45, 55, 55, 57, 57, 60, 60, 66, 66, 70, 76, 76, 85, 85, 88, 88, 91, 91, ... (moved to Word to flatten the table structure; 1 minute of work) 4) spotted by eye in the list the first tetrahedral number larger than 20: 680, the sum of 120 and 560. (Lucky the Wikipedia list was long enough :)) (2 extra minutes) 5) explained the answer in CodeProject (half an hour)

[yiangos]

Well, using a bit of python:

def pyr(l):
if l==0:
return 0
elif l==1:
return 1
else:
return pyr(l-1)+l*(l+1)/2

#then we brute-force
a=[pyr(i) for i in range(100)][1:]
b=[(i,j,k) for i in a
for j in a
for k in a
if (i!=j and i+j==k)]
if len(b)>0:
print b[0]

Running the above yields the result as

(120, 560, 680)

as others have already pointed out... (Edit: changed the type from general to Answer)

Φευ! Εδόμεθα υπό ρηννοσχήμων λύκων! (Alas! We're devoured by lamb-guised wolves!)

[YvesDaoust]

A Python script can do. Grow a list of tetrahedral numers; for every new number, try and find a number in the list such that when subtracted from the new number, it gives another number in the list. (This way you make sure that for every new number the two terms of the decomposition are already in the list.)

# Start with an empty list
List= {}
n= 1

while n > 0:
# Try the next tetrahedral number
r= n * (n + 1) * (n + 2) / 6

# Try every number in the list
for p in List:
    # Lookup the difference between the new number and the current one
    q= r - p
    if p != q and q in List:
        print p, '+', q, '=', r

        # Flag as found
        n= -1
        break

# Not found, extend the list
List\[r\]= None
n+= 1

For efficiency of the search, the list is implemented as a dictionnary. Yields:

560 + 120 = 680