piles of cannonballs

Fusion_power · 2003-08-12, 20:38

Think about a pile of cannonballs such as you may have seen if you went to one of those civil war battlegrounds. They form a tetrahedron with all faces tringular. For example, you could have 4 or 10 or 20 cannonballs in a single tetrahedron.

Now the puzzle. I want to make two stacks (perfect tetrahedrons) each of at least 10 cannonballs such that I can then re-assemble the two stacks into one stack that is also a perfect tetrahedron. What is the smallest number of cannonballs required.

I solved this in about 30 minutes!

Fusion

p.s. there are an infinite number of solutions, my way to solve it was to write a 16 line program that tried each possible combination until it found a solution.

Wacky · 2003-08-12, 22:45

I solved it in a matter of seconds. You gave too many hints.

trif · 2003-08-12, 22:50

Didn't we have this one before?

Fusion_power · 2003-08-13, 01:26

Just realized but the starting tetrahedrons must be 20 or more cannonballs each. Sorry for the miscue. Typo's happen, expecially when I'm distracted.

So if you thought you had a simple solution of 10 + 10 = 20, try again.

Fusion

ops:

Maybeso · 2003-08-13, 20:53

Quote:

Originally Posted by Fusion_power

p.s. there are an infinite number of solutions, my way to solve it was to write a 16 line program that tried each possible combination until it found a solution.

I PMed my 3 solutions, then discovered that it has been proven there are only 5 solutions - 1, 10, and the three I found. I will post a reference after you make the answer public.

Maybeso

kwstone · 2003-08-14, 02:30

Can't the two stacks be of different sizes? That would give more solutions.

cheesehead · 2003-08-14, 06:48

I don't see any requirement that the two initial stacks be of equal size.

Fusion_power's original statement, "... make two stacks (perfect tetrahedrons) each of at least 10 cannonballs ...", doesn't say the two stacks have to be the same size as each other, only that each be at least 10. Then his later amendment, "... the starting tetrahedrons must be 20 or more cannonballs each", again does not say anything about the stacks' being the same size, just that each be at least 20.

- - -

Anyway, there's a solution with unequal initial stacks that's easily found by examination of the appropriate number sequence, available on OEIS.

Jwb52z · 2003-08-14, 15:38

How can yo make a tetrahedron with triangular flat faces out of ROUND cannonballs?

Fusion_power · 2003-08-14, 19:20

How to make a tetrahedron of round cannonballs, easy, go get 10 tennis balls and stack them up. They will stack into a tetrahedron.

I received one message with the correct answer and know a few others have also found the solution. My program only tests through about 200,000 so I have to say that I don't know for sure if there are an infinite number of possible solutions. The first and obvious solution is 10+10=20. It is the only solution I found that involves equal numbers of cannonballs. Here are 4 solutions, add the pairs together to get the total count of the larger stack.

120:560 (total is 680)
1540:27720
4960:29260
10660:59640

One comment made to me was that brute force was the only way to solve this problem - i.e. there is no simple elegant formula that will tell the answer. I would agree, but after all, what are computers for if not to do brute force solutions. In a manner of speaking, that is what gimps is doing.

Fusion

Maybeso · 2003-08-14, 19:29

Actually, 10 is the only size which allows the initial stacks to be the same size. All three of the larger solutions have different sized initial stacks, and there are no other solutions, period.

Basically, you have to find a large pyramid that you can remove the base and stack it into a smaller pyramid. This gives you pyramids of height x, y, and y + 1, where the x high pyramid contains the same number of cannonballs as the base of the y + 1 high pyramid.

Wacky · 2003-08-14, 20:14

I'm not saying that there are any, but could you not also have solutions where one smaller pyramid makes up the bottom 2 layers of the large pyramid? Or 3, or more?

2003-08-12, 20:38	#1
Fusion_power Aug 2003 Snicker, AL 7·137 Posts	piles of cannonballs Think about a pile of cannonballs such as you may have seen if you went to one of those civil war battlegrounds. They form a tetrahedron with all faces tringular. For example, you could have 4 or 10 or 20 cannonballs in a single tetrahedron. Now the puzzle. I want to make two stacks (perfect tetrahedrons) each of at least 10 cannonballs such that I can then re-assemble the two stacks into one stack that is also a perfect tetrahedron. What is the smallest number of cannonballs required. I solved this in about 30 minutes! Fusion p.s. there are an infinite number of solutions, my way to solve it was to write a 16 line program that tried each possible combination until it found a solution.

2003-08-13, 01:26	#4
Fusion_power Aug 2003 Snicker, AL 1110111111₂ Posts	typo in the above, sorry Just realized but the starting tetrahedrons must be 20 or more cannonballs each. Sorry for the miscue. Typo's happen, expecially when I'm distracted. So if you thought you had a simple solution of 10 + 10 = 20, try again. Fusion ops:

2003-08-12, 22:45	#2
Wacky Jun 2003 The Texas Hill Country 1089₁₀ Posts	I solved it in a matter of seconds. You gave too many hints.

2003-08-12, 22:50	#3
trif Aug 2002 2·101 Posts	Didn't we have this one before?

2003-08-14, 02:30	#6
kwstone Jun 2003 Shanghai, China 109 Posts	Can't the two stacks be of different sizes? That would give more solutions.

2003-08-14, 06:48	#7
cheesehead "Richard B. Woods" Aug 2002 Wisconsin USA 7692₁₀ Posts	I don't see any requirement that the two initial stacks be of equal size. Fusion_power's original statement, "... make two stacks (perfect tetrahedrons) each of at least 10 cannonballs ...", doesn't say the two stacks have to be the same size as each other, only that each be at least 10. Then his later amendment, "... the starting tetrahedrons must be 20 or more cannonballs each", again does not say anything about the stacks' being the same size, just that each be at least 20. - - - Anyway, there's a solution with unequal initial stacks that's easily found by examination of the appropriate number sequence, available on OEIS.

2003-08-14, 15:38	#8
Jwb52z Sep 2002 2×3×7×19 Posts	How can yo make a tetrahedron with triangular flat faces out of ROUND cannonballs?

2003-08-14, 19:20	#9
Fusion_power Aug 2003 Snicker, AL 7·137 Posts	How to make a tetrahedron of round cannonballs, easy, go get 10 tennis balls and stack them up. They will stack into a tetrahedron. I received one message with the correct answer and know a few others have also found the solution. My program only tests through about 200,000 so I have to say that I don't know for sure if there are an infinite number of possible solutions. The first and obvious solution is 10+10=20. It is the only solution I found that involves equal numbers of cannonballs. Here are 4 solutions, add the pairs together to get the total count of the larger stack. 120:560 (total is 680) 1540:27720 4960:29260 10660:59640 One comment made to me was that brute force was the only way to solve this problem - i.e. there is no simple elegant formula that will tell the answer. I would agree, but after all, what are computers for if not to do brute force solutions. In a manner of speaking, that is what gimps is doing. Fusion

2003-08-14, 19:29	#10
Maybeso Aug 2002 Portland, OR USA 2×137 Posts	Actually, 10 is the only size which allows the initial stacks to be the same size. All three of the larger solutions have different sized initial stacks, and there are no other solutions, period. Basically, you have to find a large pyramid that you can remove the base and stack it into a smaller pyramid. This gives you pyramids of height x, y, and y + 1, where the x high pyramid contains the same number of cannonballs as the base of the y + 1 high pyramid.

2003-08-14, 20:14	#11
Wacky Jun 2003 The Texas Hill Country 3²×11² Posts	I'm not saying that there are any, but could you not also have solutions where one smaller pyramid makes up the bottom 2 layers of the large pyramid? Or 3, or more?