Deducing hidden die

Question

A dice deduction game uses a set of $N$ $(N-1)$-sided fair dice. For the $n$-th die in the set, if $n<N$, the face that shows $n$ pips is replaced with a blank. For $n=N$, all faces stays as it is.

To clarify, for $N=7$, the number of pips on each face of each dice is as follows:

Die#1: X 2 3 4 5 6 
Die#2: 1 X 3 4 5 6
Die#3: 1 2 X 4 5 6
Die#4: 1 2 3 X 5 6
Die#5: 1 2 3 4 X 6
Die#6: 1 2 3 4 5 X
Die#7: 1 2 3 4 5 6

Before the game begins, the game master hides one random die and rolls the rest of the dice on the table as the game begins. The goal of the game is to deduce which die is hidden, using this rules:

1. Player is only allowed to look at the dice when all dice have settled on their resting positions
2. Player is only allowed to look at the dice directly from above so that only the faces facing upwards can be seen
3. Player is free to move dice around as long as their orientations remain unchanged
4. Player is not allowed to pickup and inspect the dice (The point is, player is strictly prohibited from seeing any dice showing more than one face at any given time)
5. Player is free to reroll any subset of dice infinitely many times
6. Player is allowed to put mark on faces using marker as a way to carry information for future rolls

Given the rules, is there a strategy for the player to always be able to deduce the hidden die in finite time?

Answer 1

One relatively naive way that somewhat abuses rule 5 would be to roll all of the available dice repeatedly some large number (say 1000) times. At each toss of all the dice, count the number of occurrences of each face value and add it to a running total for each value. For instance, if on some given toss of the dice, 3 of them show 1, and up till now you've seen some number $N_1$ of ones total, then update $N_1 = N_1 + 3$ (forgive the computer code notation). Once you've done all your tosses, and have all your final counts $(N_1,...,N_6)$, look at the count values relative to each other. If some single count $N_x$ is significantly higher than the others, then it was probably die #$x$ that got taken out. Otherwise, if all the counts are relatively even, it is probably die #7 missing.

This works asymptotically (as the number of re-rolls goes to $\infty$) since, if the die missing face $x$ is removed, then the remaining group of dice has more $x$ faces between them than any other value. And if the die with all its faces is removed, there is an equal number of each face value left in the group.

The success rate of your game will of course be a function of how many re-rolls you do, and the precise form of rule you use to decide whether one count is "significantly higher" than the others.

Edit: You could also further abuse rule 5 by instead taking each available die individually, rolling it some number of times where you'd expect to see every side at least once (say 30 times). Then decide for that single die which one you think it is (based on which face value(s) you saw zero times) and set it aside. After you do that for all available dice, 1 should be missing from the list of "identified" dice. This seems like it might be even faster than the first way I gave. Basically, rule 5 leaves you with a few different ways to go about it.

Of course neither of these methods gives success with certainty, but if you use good "decision rules" then there should be success with probability approaching 1 as the number of re-rolls increases.