Eliminating random numbers as they're generated

You have 2 options.

Option 1; Deck of cards.
The easy one is to have a deck of cards. Once a card is picked, it is excluded from the deck.
But I can figure that 80 cards is a bit much.

***

Option 2; Dice.
The tricky one is to use dice any way. But no re-rolling.
This goes by counting. What I mean is the following:

Let's pretend you have a grid of 10 tiles.
They are not numbered. But you simply start counting from a certain direction. From left to right. The 10th tile is at the end.

1234567890

You use a d10 for all the rolls.

You roll a 5?
You count from left to right until you reach the 5.
The 5th tile is picked and becomes something called A.

1234A67890

But now you want to roll again for the next tile.
You need to recount. So the grid actually should be like this in your mind:

1234A56789

If you are to roll a 5 again.
You have a new 5th tile.
Which was previously the 6th tile.

You now get:
1234AB6789 or
1234AB5678

As you can see, the last number (that you count in your mind) gets lower every time.

If you where to roll a 10 now.
What happens next?

You simply count to 8.
Then you have 2 left.
You have several choices now to decide on:

- The players count again, starting at the 1st tile.
They end up on tile 2.
- The players count backwards, starting at tile 8.
They end up on tile 7.
- The players count backwards, starting at tile 7.
They end up on tile 6.

I can't tell you which one to pick.
All 3 have the same effect.
All second counted tiles will have a double chance on being picked.
But if that is ok with you, this is the way to go.

Personally, I would go with the second option. So that the most outer tiles aren't picked only once.
The first option is just as good.
But gives less randomness feeling (yeah, it is only the feeling here).

You always roll once for a tile.
You always can pick a tile.

***

With a grid of 80. You can't use the technique on both dice.
I suggest to use a d8 and d10.
d8 for top-down (I think)

d8 is rolled to determine the line of tiles.
You can only exclude a line of tiles, if they are all used up.
Once d8 has determined the line of tiles, you look at your d10.

Another possible solution. Preprint a list of a thousand sets of numbers that match your criteria and number them. Then roll d10 four times. Write down the results of the rolls (10=0) and then use that number to determine which set of numbers to use.

If you are familiar with computer programming think of it accessing an array using a randomly generated index.

This is a hard task to do it manually: getting 80 random numbers with dice is not so easy and, any solution, is far to be optimal.

I think a better idea is just use an standard deck of 52 cards, this seems more practical and easier. If you need to determine 80 numbers you can select 40 cards of the 52 that are easier to assign a value, and repeat after the same procedure for the other 40 positions.

Since your game board are divided up into groups of 20, create 20 unique, numbered tiles. Mix them face-down, and flip them up in a row. Roll a die (d6 would be fine) and count that many along that row, and the number drawn is the number used. Remove that tile from the row, sliding all tiles to the left to fill in empty spaces, and then roll the die again.

Repeat until all tiles have been sorted. To fill in another quadrant of 20 spaces, mix all the tiles and repeat the process.

Might this accomplish your task, and use fewer components?

Role a d20. Position the closest vertex of the rolled number towards you. Adjust your view so that you only see the five faces attached to this vertex. It should look like a pentagon with five triangles that meet in the middle. Starting with the number rolled record each number in a clockwise fashion. On the fifth number look at the connecting number that is not used. Position the original vertex to be the north pole. Then record this number and rotate the die clockwise in an record each number in the "equator". On the 10th "equatorial" number place the adjacent number that has not been used on top with the vertex opposite the shared edge closest to you. Again this should now look like a pentagon with five triangles that meet in the middle. Record this number and each number in a clockwise fashion.

Admittedly this only produces 60 results. If that is not good enough. You could add more to this with randomizing whether to clockwise or counter-clockwise. (I choose clockwise because it was shorter to type.)

If the players need to pick and choose certain locations to go into the process each turn, chits would probably be more durable.