A pair of dice which never roll 7 (2004) | Hacker News Reader

A pair of dice which never roll 7 (2004)

chiark.greenend.org.uk

130 pointsajot53 comments

For a while they sold a pack of 36 cards as a Catan accessory, labeled with 2 through 12 in the same frequencies as two six sided dice, and instead of rolling the dice you flip over the top card and then you shuffle them and start over when you reach the bottom.

It's an interesting change to gameplay, since the numbers are no longer independent and the gambler's fallacy ("I haven't had a 12 for a while, so it's more likely to come soon") actually makes sense here. You could restore independent numbers by reshuffling the entire deck after each turn.

That seems like an easy way to get the results from the article, without worrying about biased dice or repainting. If you want to alter the deck's probability you can just add or remove cards.

O that's really interesting. I've been thinking about making something like that myself, didn't know there was an official extention.

I think it should improve the game when you remove a bit of luck with this.

The dice deck is alright, but just alright. Even if you include the card that tells you to reshuffle the deck, the bell curve is just too smooth. If you have a roll drought, you know that deluge is on its way.

The best thing I have done for dice in my Catan set is to include a bag with a random assortment of 6 sided dice. At the beginning of a new game we randomly pick two dice. The random sizes, materials, and origins of manufacture keep the clustering of rolls inconsistent from game to game, which makes the game more enjoyable.

Catan universe has a “card stack” mode where they take two sets of those 36 card decks and randomly remove 18. You end up with some randomness but much better bell curves on dice distribution. My friends and I debated for a while but once we switched to using card stack we never went back.

> re-rolling a pair of dice until you get a result you like seemed ugly.

... then proceed to:

- create an entire new pair of dice for 2 turns of a game they play once in a while

- one die now has 2 values on each face with separate meaning

- for the first 2 turns, you have a special procedure to read the value of the dice that is quite convoluted

I understand the fun of it, but that's also how I see code being written in the corporate world, and that makes me sad.

> the dices

"The dice". Single "die", plural "dice".

French here. Thank you.

Just to further complicate things.... native English speakers often use "dice" as the singular i.e. instead of "die". While technically incorrect, it's very common, probably more common than the "correct' form.

The plural is "dice" and unfortunately "dices" is definitely incorrect :-)

English is crazy.

I’ve even heard people use “die” as both the singular and plural! It’s the only singular/plural pair I know of where 3 of the 4 permutations show up in common parlance. I’ll be thrilled if I ever hear someone use dice as sing. and die as pl.

Some constructs in English can indeed feel a bit dicey.

Ah, you pipped me to it.

The plural of inconsistency:

https://diaspora.glasswings.com/posts/897e437031880139f09900...

Mouse, mice therefore singular of dice is douse!

I think the word itself is from French too (although I don't know what the explanation for the strange plural form is).

From the OED: "As in pence, the plural s retains its original breath sound, probably because these words were not felt as ordinary plurals, but as collective words".

Here is another interesting article about the etymology and singular/plural usage: https://www.grammarphobia.com/blog/2016/10/dice.html

Doesn't the word dies connote exactly two dies? Then with three or more it becomes dice.

No.

The pair-of-dice / paradise pun would be burned into your brain if you spent more time with bikers.

I don't think English has any plural form for >=3.

A pair of dies is dice! But then what does 'pair of dice' mean?

Which brings us to another problem: I can buy a new pair of pants - or even 'some new pants' - but I've never seen 'a pant' for sale !! ;-)

The red numbers did seem like stretching it too far, but I actually think the original construction is quite elegant. Taking the number with more dots doesn’t seem like any more mental overhead than mentally adding the values on two normal dice.

Since he already takes the one with a higher number of dots on either die, wouldn’t that make it a d12?

No, because a d12 has a uniform probability distribution across all numbers which is explicitly not what is wanted for Catan

This is a very tricky problem. We want dice that generate outcomes of two fair dice but no 7s, and every other outcome having the same relative probabilities. Note the generating function (x + x^2 + x^3 + x^4 + x^5 + x^6)^2 - 6x^7 has no nontrivial factors, so we must get creative. My first solution was to consider the generating function x^2 + 2x^3 + 3x^4 + 4x^5 + 5x^6 + 5x^7 + 4x^8 + 3x^9 + 2x^10 + x^11 = (x^6 + x^5 + x^4 + x^3 + x^2 + x) (x^5 + x^4 + x^3 + x^2 + x). So we can do it with a standard 6-sided die and a 20-sided die with 4 each of 1,2,3,4,5. If you roll a 7 or more, add 1. My solution is completely different from anything mentioned in the article. I actually like my solution much more, as it uses two readily available Platonic solids. You don't even need to renumber! For the D20, you can divide the face number by 4 and ceiling, or mod 5 and add 1.

There's a more obvious interpretation here. After you've rolled one die, there is exactly one other roll for the second die that could produce a 7. If you remove that possibility, you have a d5, still numbered 1-6 but with one of those digits missing.

Excellent point!

Wait, let me see if I got this right... I use a Red die and a Blue Die. Blue die is considered "secondary" (in the examples below, the second number is always taken from the blue die), and both dice are thrown together.

- I roll 4+2 - result is 6

- I roll 4+6 - result is 10

- I roll 1+6 - result is 1

- I roll 3+4 - result is 3

- I roll 4+3 - result is 4

Correct?

Wouldn’t that change the probabilities though?

This is what I suspect, too, and I am too lazy to try this out on Anydice anyway, so I asked to be sure I am not missing something.

But this messes with the distribution (you now have two ways of rolling a 2: 1,1 and 2,5) and makes outcomes like '1' possible (1, 6), which shouldn't be

a normal D6 and a D10 with 1-5 twice. If total is 7, use the D6 plus 6 instead. The roll table for that is exactly that of 2 D6 with the 7s removed. The 7s shift to the last column.

      1  2  3  4  5    <- D10
    1 2  3  4  5  6
    2 3  4  5  6  >  8
    3 4  5  6  >  8  9
    4 5  6  >  8  9  10
    5 6  >  8  9  10 11
    6 >  8  9  10 11 12

Oh I see, I misunderstood. Yep I like your way best so far although I didn't see the problem with rerolling

My simple solution was:

- Use a normal d5 and d6.

- On a roll, if the sum on the dice face is >= 7, add 1 to the sum.

If I roll 2+3, the result is 5. If I roll 3+4, the sum is 7, so add 1 to get the result of 8.

The gives you this distribution, and is simple in practice.

      1  2  3  4  5
    1 2  3  4  5  6
    2 3  4  5  6  8
    3 4  5  6  8  9
    4 5  6  8  9  10
    5 6  8  9  10 11
    6 8  9  10 11 12

One thread from long ago:

A Pair Of Dice Which Never Roll 7 - https://news.ycombinator.com/item?id=292070 - Sept 2008 (9 comments)

You could also allow some labels to be negative. For instances, dice (3, 4, 4, 9, 10, 10) and (-1, 0, 1, 1, 1, 2) result in all numbers from 2 to 12 without 7, but obviously not with the same outcome distribution. It's also not possible without repeating some labels.

Your negatives aren't necessarily. You can subtract 2 from everything on the first die and add 2 to everything on the second die.

Yes indeed, I hadn't noticed that.

I had some fun with something similar by printing a 6-sided die with one of the numbers appearing twice, and one missing. If one is not looking for it it is very hard to spot randomly. I had the kids roll the trick and store-bought dice some 200 times, and record the rolls on a chart. At some point it clicked :) Also a good demo of randomness and what it actually looks like in practice to be "fair" vs. "not fair".

The board game Glow kind of does this. There are only five symbols of interest which allows for having color coded dice with an additional side of that color (eg red dice have an additional fire symbol).

Beautiful artwork, 4/5 game play, definitely recommended.

>Now the procedure is: roll both dice. To find the total, choose whichever of the two uppermost faces shows more dots. To find the red number, take the red number shown on the even (R) die, and modify it by the amount shown on the odd (C) die if any. If an addition takes you above 6, subtract 6 from the result; if a subtraction takes you below 1, add 6. [...] The mechanism I've described above works nicely on paper, but unfortunately it has a practical disadvantage, which is that it's unusually sensitive to imperfections in the dice.

While this is interesting, it hardly seems like potential nonuniformity due to unbalanced dice is the big issue here. I'd consider this an unusually complicated and awkward dice counting mechanic in a ttrpg. For a casual boardgame, it is beyond the pale.

I agree, cool problem solving, but ultimately a contrived solution. I found the next bit after your quote quite funny after thinking of this solution right away while reading.

>A really obvious alternative approach would have been to roll a d6 (labelled 1-6 as usual) and a d5 (also labelled 1-5 in the obvious way), add the results, and then add one if it was 7 or more. . . I initially discarded this approach because it felt inelegant; I wanted my dice to roll (say) a 9 by producing something which was obviously a 9, rather than producing something that looked like an 8 and requiring you to remember to convert it into a 9.

Because one simple rule is so inelegant compared to dice with numbers, dots and sometimes a second number, riiight.

Yeah I mean if you want elegant just reroll the fuckin dice. The solution is kind of interesting I guess, but it's so impractical I don't think you can even call it a solution.

It's a colloid.

Geez, what happened to the hacker spirit? The guy saw a specific problem and thought it an interesting challenge to tackle. He showed that it's -- in theory -- possible to do it. And that's an obvious results, given that (a) he still wanted it to be two dice, like in the original; and (b) the relative probabilities of the other outcomes should remain unchanged.

I think it's cool.

If this is just for the sake of not rolling 7 you can have a die with just odd numbers like 1, 3, 5, 7, 9, 11. Having 2 of them guarantees an even sum. Of course you would have gaps between the possible outcomes but not rolling a 7 is already a gap.

But it's not just for the sake of not rolling 7, it's about avoiding only that number to improve Catan. If you can't roll any odd number in the first few rounds, tiles with those number become way less valuable, which changes where you want to place your villages, which affects the entire game.

The 7s can be replaced with 0s to fill in some of the gaps and allow for odd number rolls.

Why make face 7 blank? Why not keep 0 dots under it to make it the lowest priority? Would that mess up the probabilities?

Yes. You would have to re-roll the C dye until not 7. Spared re-rolling both dice though.

Why not simply take the dice result not literally, but as index to a mapping table there 2 or 12 is swapped with 7?

Or use 1 dice 1..6 and one dice 5..10

The result should be (excel syntax) mod(d1+d2;12)+1. You can adjust the lowest start number of dice 2 to control which result gets the highest probability

that would change the probability distribution of the remaining numbers, which is to be avoided

I thought I was going to read some article that explained some sort of Stuff Made Here type electronically loaded dice that communicated with each other to avoid landing in positions that added up to 7.

The mathematical approach is more elegant, but less hacker

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