Here are some ideas to play with variations of this game. You can treat these variants as modules that can be combined as well.
1. A longer version with 6 colors
The gameplay is exactly the same, except for two things:
A) There are 2 additional colors on which to score, so you will need 2 extra colors of dice.
B) The additional colors have numbers in another order than the basic order from the 'normal colors'.
The sheet will now look something like this:
Red: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, lock
Yellow: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, lock
Green: 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, lock
Blue: 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, lock
Pink: 2, 12, 3, 11, 4, 10, 5, 9, 6, 8, 7, lock
Orange: 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, lock
Now there are more possibilities to lock a color with the neutral dice.
2. The grey die
An extra (large) grey die is added. This die works much like a white die, but with one important difference. It must always be rolled with the other dice, but it can only be used on another player's turn and you must combine it with a colored die. Combining it with the neutral dice is not allowed.
3. The purple die
An extra (large) purple die is added. It must always be rolled with the other dice. If the result of the purple die matches one of the other dice, including the white dice, then that die is 'blocked' in that turn. This means that the result of that die may not be used in any way during turn by neither the active player nor the other players. It only counts against one of the other dice. If it matches more than one of the colored dice, then the active player may choose which die to block. However, if one of the dice it matches with is white, then the white die will always be the die that is blocked. Note: if the grey die is added as well, the colored die that is blocked can not be used by another player.
At the start of the next player's turn, dice are no longer blocked by the purple die.
4. The brown die
An extra (large) brown die is added. If a row is locked, the die of that row's color is NOT removed from the game. It is optional to roll the brown die. If a brown 1 is rolled, then you can choose to unlock the row of a color in which you can still mark some numbers. However, you can only do this if it is phase 2 of your turn (colored combination) and the combination of the other dice would allow you to mark an eligable number as if the row was not locked. You may only unlock one row this way and you may only unlock it if the colored combination allows you to mark an eligable number after opening that row. You may not unlock one row and then choose to combine the colored die of another row to mark a number or not mark a number at all. The white dice roll in phase 1 of the turn does not count, so you may not unlock a row to mark a number with the neutral roll. The player that locked the row can never unlock it, because he can not mark an eligable number in that row. All rolls other than a 1 on the brown die have no effect.
Example: The red row has been locked by one of my opponents on his turn. The blue row is still open. On my score sheet, the best eligable number in the red row to markis a 9 and in the blue row it is a 6. Usually it is not allowed to mark the 9, while I have very few marks in that row. So I choose to throw the brown die together with the others. The result of the dice include a white 3, a white 2, a red 6, a blue 4 and a brown 1. Usually, I'm unable to mark the 9 with the white 3 and red 6, because the row is locked. But I'm lucky to roll a brown 1, which allows me to unlock the red row and mark the 9 after all. On a later turn I can attempt to mark the other numbers as well, unless the row is locked again. I can also choose to mark the blue 6 using the other white die, but then I'm not allowed to unlock the red row. I may not unlock the red row and choose not to mark the 9. If I want to unlock the row, I must mark the 9 as well.
5. The black die
An extra (large) black die is added. It is optional to roll it on your turn and if you do, you roll it together with the other dice. The ability of the black die can only take place at phase 2 of your turn and is optionally resolved on a roll of a black 5. If a black 5 is shown, the active player checks the result of the other dice. If he can mark an eligable number in phase 2 of his turn, and he does, then he will have the option to choose another player. That player's row of the color in which the active player marked an eligable number is 'banned'. This means that it is locked, but this only counts for that player and that player can not mark the lock as a consequence of this effect. If a player's row is banned at the start of his turn and he rolls a black 6, then he may choose a row to 'unban', which means he can mark numbers in that row again. In the case the brown die is added as well, its ability does not count as unbanning a row after it was banned by the black die. The purple die, if it is added as well, can cancel the effect of the black die, when a black 5 is rolled, if it matches the colored die of the row that the active player wants to ban, because he can't mark a number of that color that way.
I did not playtest this. I was just brainstorming about some ideas to make this game even more fun to play. Feel free to give some feedback.
I like 1) and 2).
I haven't played the Big Numbers expansion, but 1) is more of what I was hoping they would do. It seems an obvious change to make rows that are something like 7,8,9,10,11,12,2,3,4,5,6; easy to start an end, but you have to decide whether to wait for the low probability middle numbers to rolls. I think I would add a rule that locking pink or orange don't count towards the "2 locked" end condition.
2) is an easy variant to test, and gives more decisions for players when it's not their turn. I would make explicit that a player must decide to use or not use the total of the 2 white dice before then using the total of grey+color, just as the active player has to.
A couple variants I was considering were:
The direction of the numbers are flipped for different players, with the 6 possible combinations.
So one player has: Red 2-12, Yellow 12-2, Green 12-2, Blue 2-12
Another: Red 12-2, Yellow 2-12, Green 2-12, Blue 12-2
And so on.
Polyhedrals: replace the red and blue with d8. And then change red and blue rows on the sheet to:
For these 6 spots are needed before you can claim 14 or 2 and lock.
There's lot of different things you could do with polyhedrals and non-standard numbering to change the probabilities while still restricting the range from 2-12 and not requiring new sheets. For example, a nearly flat probability could be produced by using a red d6 numbered 1,1,1,6,6,6