There's always a mathematical puzzle in the Saturday times which I like to have a go at.
This weeks is about making towers out of standard dice.
Essentially the question is how many towers of equal height can be constructed so that the visible pip count equals exactly 700.
Well I couldn't work iit out so I looked at the answer which was six towers of eight dice each.
Now my logic is this:-
For the first 7 dice in the tower the visible pips are 14(all opposite sides add up to seven.)
For the eighth dice the total is 20(14 plus a 6 on the top)
7*14 + 20 =118.
118 *6 = 708 which is not "exactly" 700.
Am I missing something?
@venda saidThinking about it again I think I have sussed the "trick"
There's always a mathematical puzzle in the Saturday times which I like to have a go at.
This weeks is about making towers out of standard dice.
Essentially the question is how many towers of equal height can be constructed so that the visible pip count equals exactly 700.
Well I couldn't work iit out so I looked at the answer which was six towers of eight dice each.
Now my ...[text shortened]... a 6 on the top)
7*14 + 20 =118.
118 *6 = 708 which is not "exactly" 700.
Am I missing something?
The top die on each tower doesn't have to show a six.
Also the top die on each tower could show a different number..
That makes it very complicated.
The formula must be something like (14x+y)*z =700
Where x is the number of die -1, y is the number of pips showing on the top dice and z is the number of columns.
How you work that out mathematically when y and z are both unknowns is beyond me
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@venda saidYou're assuming that each tower must have the same number showing up top. That, too, is not a given from what you've written.
Thinking about it again I think I have sussed the "trick"
The top die on each tower doesn't have to show a six.
Also the top die on each tower could show a different number..
That makes it very complicated.
The formula must be something like (14x+y)*z =700
Where x is the number of die -1, y is the number of pips showing on the top dice and z is the number of columns.
How you work that out mathematically when y and z are both unknowns is beyond me
I rather think you need to post the literal text of the puzzle for us to be able to solve anything. Right now, it's like trying to find out why a program crashed when you only have the code for half the functions. And we're not Stack Overflow, so we can't just randomly guess.
@shallow-blue saidYes I already said that I had wrongly assumed the top die was showing a six on the top.
You're assuming that each tower must have the same number showing up top. That, too, is not a given from what you've written.
I rather think you need to post the literal text of the puzzle for us to be able to solve anything. Right now, it's like trying to find out why a program crashed when you only have the code for half the functions. And we're not Stack Overflow, so we can't just randomly guess.
I have also given the answer.
The question is,is there a way of calculating the answer without resorting to "trial and error".
However for completeness:-
The queen of Nootropia stacked some dice one on top of the other to make a tower.
She then made some more of the same height.
Afterwards the queen surveyed her creation,counting all the visible pips.
There were 700 pips in total.
She noticed these were the tallest towers she could have built with precisely 700 visible pips.
Q:-how many towers had she made?
Altough it doesn't say so, I think we must assume there are the same number of die in each tower(they could be the same height if for example one die in one of the towers was twice as big as all the others)
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@venda saidThe tallest! Yeah, that'll make a critical difference to the puzzle, all right...In your first post, you wrote "how many towers ... can be constructed", which might lead you to assume it's the number of the towers that needs to be maximalised, not their height.
She noticed these were the tallest towers she could have built with precisely 700 visible pips.
Q:-how many towers had she made?
The answer to the problem in your first post is 36 towers of one die with the one on top, plus 10 with the two on top. Total 46 towers and 46 dice. This can be found easily by dividing 700 by 15 (the minimum number of pips for a one-die tower) and noting that 47 towers would have at least 705 pips.
The real problem is more complicated - and more interesting! The total number of pips shown on t towers of d dice each is anywhete between 14×t×d + 1×t and 14×t×d + 6×t = 700. Maximise d; therefore, minimise t.
Note that 14 divides 700. If we could have zero pips showing on top, the problem would be simple (and the answer would be one tower of 50 dice). If we counted both top and bottom faces, it would be almost as simple: four towers of twelve dice each. No, not two of 24.
But it's not that simple...
@eladar saidIt means all sides plus the top number on the top die
How big are the dice? If they are 1 meter by 1 meter by 1 meter, you could never see the top.
Does visible mean by walking around? Or just one perspective, which allows you to see 1 side, perhaps 2.
@venda
The key is considering the top faces.
Let n = number of towers.
Let h = height towers
Let X = total of top faces
Considering top faces X < 6*N
Then we have (14*n*h) + T = 700
(Because 700 is divisible by 14 then necessarily X must be too.)
Let T = 14*X
Then (n*h) +X = 50 with the constraint that 14*X < 6*n
A little leg work now.
Trying X=1 gives 7 towers, 7 dice high. (possibility since 14 < 42)
Trying X=2 gives 4 towers, 12 dice high. (impossible because 28 > 24)
Trying X=2 gives 6 towers, 8 dice high. (possibility because 28 < 36)
Trying X=3 gives no solution
Trying X=4 gives no solution
Trying X=5 gives 5 towers, 9 dice high. (impossible because 126 > 30)
and one can see that as X increases there are no solutions.
Hence X=2 gives us the answer.
Here's my line of thought for the solution. It is similar to wolfgang's in some ways.
There are always 14 spots showing on every die but the top ones. 700/14=50, BUT we must add at least one spot on top, so we go over 700. Therefore, the total number of dice used must be less than 50.
This means we must take away at least one die [losing 14 spots per die] and then arrange the top spots to get back at least as many spots as we lost. The max amount of spots we can gain back is 6 times the number of stacks.
To ever reach 700, we must have at a minimum 3 stacks, because 6*3 = 18, which is greater than the minimum loss of 14. In general, the number of stacks must be at least 3 times greater than the number of dice removed.
Since we want to minimize the number of stacks, and also minimize the number of dice removed, we want to shoot for
dice_removed * 3 = n_stacks
Since we can't get that with 49 dice, 48 must be optimal [the 3 is a factor of it]. This removes 2 dice, so we must have 6 stacks.