a puzzle

bcp19 · 2012-02-20, 22:16

You have 680 glasses and need to build a tower 15 levels high with 1 on top. Is there an equation that can be used to solve for glasses per level?

science_man_88 · 2012-02-20, 22:21

Quote:

Originally Posted by bcp19

You have 680 glasses and need to build a tower 15 levels high with 1 on top. Is there an equation that can be used to solve for glasses per level?

only thing that comes to mind is (680-1)/14 =48.5 / level on average.

bcp19 · 2012-02-20, 22:37

Quote:

Originally Posted by science_man_88

only thing that comes to mind is (680-1)/14 =48.5 / level on average.

it's more like a pyramid, your design would not be stable.

chalsall · 2012-02-20, 22:41

Quote:

Originally Posted by bcp19

it's more like a pyramid, your design would not be stable.

You didn't define stability as being a requirement.

bcp19 · 2012-02-20, 22:48

Quote:

Originally Posted by chalsall

You didn't define stability as being a requirement.

True, but that equation would be simple to figure out, plus how would you have 1 on top and have that .5 per level?

Dubslow · 2012-02-20, 22:50

Well the sum over the integers up to n := the nth triangular number would describe perfect triangles. So find the highest triangular number <= 680, then distribute the remainder evenly over each layer. In C:

Code:

int * pyramid(int x) {
        int i,j,diff;
        int sum=0;
        int * layer;
        printf("Input is %d\n", x);
        for( i=1; sum <= x; i++)
                sum += i;
        sum -= (--i); i--;                                                      // The loop goes one too far, so fix it
        layer = (int *)malloc(i*sizeof(int));                                   // Declares an array of size i
        for( j=0; j<i; j++)
                layer[j] = j+1;                                                 // The basic shape is a triangle
        diff = x - sum;
        if( diff > 0 ) {
                for( j=i-1; diff>0; diff-- /* Decrement diff until diff==0 */ ) {               // Distribute the difference over each layer
                        if( j<1 ) j = i - 1;                                    // Keep cycling over the pyramid
                        layer[j]++;                                             // Add one to the current layer
                        j--;                                                    // Move to the next higher layer
                }
                printf("There will be %d layers, as follows:\n", i);
                for( j=0; j<i; j++) printf("%d\n", layer[j]);
                printf("\n");
                return layer;
        }
        else if( diff==0 ) {printf("A perfect triangle of glasses!\n\n"); return NULL;}
        else {printf("You need %d more debugging printfs!\n", x); return (int *)"YOU SUCK";}
}

Will post results for 680 in a moment (if other inputs give the right outputs). Edit: For pyramid(6), it gave me 1,2,3,4,5 so something's not right. Hang on. Edit: Found one bug. The sum goes too far. (Not done yet though.) Edit: Got em all. Also realized that this isn't exactly optimal, but it's a solution.

Code:

What number of glasses should I use?
1
Input is 1
A perfect triangle of glasses!

What number of glasses should I use?
2
Input is 2
There will be 1 layers, as follows:
2

What number of glasses should I use?
3
Input is 3
A perfect triangle of glasses!

What number of glasses should I use?
4
Input is 4
There will be 2 layers, as follows:
1
3

What number of glasses should I use?
5
Input is 5
There will be 2 layers, as follows:
1
4

What number of glasses should I use?
6
Input is 6
A perfect triangle of glasses!

What number of glasses should I use?
7
Input is 7
There will be 3 layers, as follows:
1
2
4

What number of glasses should I use?
8
Input is 8
There will be 3 layers, as follows:
1
3
4

What number of glasses should I use?
680
Input is 680
There will be 36 layers, as follows:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
24
25
26
27
28
29
30
31
32
33
34
35
36
37

When I say not optimal, I mean it seems to me that a stack of 5 glasses should be 1-2-2, not 1-4 as this code suggests, and 2 should be 1-1, not just 2. Will have to come back to this later though, I have to go code for my CS HW due in 2.5 hrs

(Somebody else want to optimize the algo?)

EDIT: WHOOPS 15 per layer. There's a HUGE piece of missing information. Well, at least I had fun solving my own problem :/ (Another edit: This is a 2-D solution, as opposed to eh 3-D ones below :P:P:P)

chalsall · 2012-02-20, 22:53

Quote:

Originally Posted by bcp19

True, but that equation would be simple to figure out, plus how would you have 1 on top and have that .5 per level?

As I have learnt the hard way... Be very careful what you ask for -- you might just get it....

science_man_88 · 2012-02-20, 22:53

Quote:

Originally Posted by Dubslow

Well the sum over the integers up to n := the nth triangular number would describe perfect triangles. So find the highest triangular number <= 680, then distribute the remainder evenly over each layer. In C:

Code:

int * pyramid(int x) {
    int i,j,diff;
    int sum=0;
    int * layer;
    for( i=0; sum <= x; i++)
        sum += i;
    layer = (int *)malloc(i*sizeof(int));    // Declares an array of size i
    for( j=0; j<i; j++)
        layer[j] = j+1;                               // The basic shape is a triangle
    diff = x - sum;
    if( diff != 0) {
        for( j=i-1; diff>0; diff-- /* Decrement diff until diff==0 */ ) {               // Distribute the difference over each layer
            if( j<1 ) j = i - 1;                        // Keep cycling over the pyramid
            layer[j]++;                                // Add one to the current layer                               
            j--;                                           // Move to the next higher layer
        }
        printf("There will be %d layers, as follows:\n", i);
        for( j=0; j<i; j++) printf("%d\n", layer[j]);
        printf("\n");
        return layer;
    else if( diff==0 ) {printf("A perfect triangle of glasses!\n"); return null;}
    else printf("You need %d more debugging printfs!\n", x);
}

Will post results for 680 in a moment (if other inputs give the right outputs).

Quote:

Originally Posted by PARI

Code:

(18:20)>sum(X=1,15,X)
%37 = 120

so it's definitely not a perfect triangle.

bcp19 · 2012-02-20, 23:00

Quote:

Originally Posted by science_man_88

so it's definitely not a perfect triangle.

Actually, that may work...
1+3+6+10+15+21+28+36+45+55+66+78+91+105+120=680

yep, that works... I was thinking 15x15,14x14, etc but it went well over 1k.

With the levels now known, without using trial and error, is there an actual equation that would solve this?

ccorn · 2012-02-20, 23:13

Quote:

Originally Posted by bcp19

You have 680 glasses and need to build a tower 15 levels high with 1 on top. Is there an equation that can be used to solve for glasses per level?

It's a pyramid with triangles at each level, each triangle packed regularly with glasses in the (2-dimensional) densest possible way, one glass at the top, and each non-top level having its side length one glass longer than the previous (higher) level.

Numbering the levels beginning with 1 at the top, the glass count for level i is the triangular number T(i) = i*(i+1)/2.
The total glass count V(n) for n levels is then
$V(n) = \sum_{i=1}^n \frac{i\,(i+1)}{2} = \frac{1}{2}\left(\left(\sum_{i=1}^n i^2\right)+\left(\sum_{i=1}^n i\right)\right) = \frac{1}{2}\left(\frac{n\,(n+1)\,(2n+1)}{6}+\frac{n\,(n+1)}{2}\right) = \frac{n\,(n+1)(n+2)}{6}$ .
And indeed V(15) = 680.

Edit: OK, folks, you have been faster.

kar_bon · 2012-02-20, 23:21

Quote:

Originally Posted by bcp19

With the levels now known, without using trial and error, is there an actual equation that would solve this?

Every level n is the sum of all numbers from 1 to n.

So:

FOR level = 1 to 15

number of glasses = $\sum_{i=1}^{level}i$

2012-02-20, 22:16	#1
bcp19 Oct 2011 1010100111₂ Posts	a puzzle You have 680 glasses and need to build a tower 15 levels high with 1 on top. Is there an equation that can be used to solve for glasses per level?

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Some puzzle	Harrywill	Puzzles	4	2017-05-03 05:10
Four 1's puzzle	Uncwilly	Puzzles	75	2012-06-12 13:42
4 4s puzzle	henryzz	Puzzles	4	2007-09-23 07:31
Dot puzzle	nibble4bits	Puzzles	37	2006-02-27 09:35
now HERE'S a puzzle.	Orgasmic Troll	Puzzles	6	2005-12-08 07:19