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i came across a "math-problem" for students in middleschool: A stack of dices is placed on top of each other and the task is to find a term that describes the number of sides that are not covered by the table or a subsequent dice.

I am not satisfied with the solution given in the textbooks and my school days are (unfortunately) too long ago to ask my old maths teacher, so I am trying my luck here ;)

I have linked the problem here (it is in German, but I am only interested in the visual representation). The solution suggested is 4x+1="number of sides".

What bothers me is that it doesn't include the case where there is no cube. So if x is 0, the result of the number of sides should also be 0.

I have created a table of values with my maths skills and entered it into wolframAlpha, but am failing to come up with a correct function that "gives" "my" solution. I hope I have expressed myself clearly and that someone can help me.

Many thanks in advance!

4 those, who do not trust links ;)

choXer
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    I think the $4n+1$ formula is meant to apply to a vertical column of $n$ dice, and your concern is that this works only when $n≥1$. Is that right? – MJD Nov 05 '21 at 23:02
  • @MJD Yes, exactly :) – choXer Nov 05 '21 at 23:03
  • Because there is a "stack" of dice, this condition indicates that $n$ must be at least $2$. In any case I do appreciate your thoroughness and I think that trait will take you far in mathematics! Note singular = "die" and plural = "dice." – Gwendolyn Anderson Nov 05 '21 at 23:09
  • @GwendolynAnderson That's actually due to the lack of my english skills. The 4x+1 Formula is the "correct" one. OT: Thanks for the note about die (thats almost horrorfying ;) ). Do you have an idea how to come up with a formula that includes n=0? – choXer Nov 05 '21 at 23:13
  • For what it's worth, when $n \geq 1$, the formula can be intuitively derived as follows. The number of visible sides is $6n$ minus the number of hidden sides. Every die except the one at the very top of the stack will have its top and bottom faces hidden, while the die at the top of the stack will only have its bottom face hidden. Therefore, when $n \geq 1$, the number of hidden faces is $(2n-1).$ Then, the number of visible faces equals $(6n) - (2n-1)$. – user2661923 Nov 06 '21 at 00:08
  • The technical term for the second formulation given by @Milten is a Bernoulli Trial, which has two outcomes, "success" (value one) or "failure" (value zero). [https://en.wikipedia.org/wiki/Bernoulli_trial] It is especially useful if you are writing computer code in which case you can subtract the trial ($n=0$) times the value which in this case is just $1$. A "failure" outcome zeros out the adjustment. So the second formulation is taking over due to programming languages while the first one is the traditional mathematically correct expression ... and the third contrivance is clever! – Gwendolyn Anderson Nov 06 '21 at 17:15

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The formula will just be $$ f(n)=\begin{cases} 0 & n=0 \\4n+1& n\ge1.\end {cases}$$ If you really want to, you can use an indicator function, for example like $$ f(n) = 4n+1-[n=0]. $$ Alternatively, you could paint a yellow square on the table so that $0$ cubes gives one visible square ;-)

Milten
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