# Questions tagged [dice]

For puzzles involving any form of dice.

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### How many tries to roll a 6?

Suppose you roll a (fair, 6-sided, perfectly ordinary) die repeatedly until you roll a 6. As is well known, the expected (i.e., long-term average over many trials) number of rolls required is 6. Now ...
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### Relabeling two 20-sided dice without changing their total

Usually, 20-sided dice are labeled with the numbers 1 to 20. When you roll two of these, their sum is a random number between 2 and 40. The total 21 is most likely to occur, while 2 and 40 are the ...
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### Aproximating 100 by 6

You're a fan of pen & paper RPGs and have finally arranged a new playing group, recruited amongst your friends from the mathematical faculty. As a GameMaster (GM) you've assumed that all your ...
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### A Boyfriend's Mysterious Message

Imogen was dog-tired, but excited! After 3 gruelling weeks of 16-hour work days imposed by an iron-willed boss, she was finally finished all her projects. She'd been processing (figurative) ...
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### Magician, assistant, nine dice, and two cups

A magician and her assistant are performing a magic show today, and you are the only audience member. Since it is a Thursday, the magician is saving her strength for the inevitable dragon attack and ...
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### Gambling in a rigged casino

A well-known crooked casino is offering the following bet with a pair of (identically looking and indistinguishable) dice that both show the numbers $1,2,3,4,5,6$ on their faces. The first die ...
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### Fair results with unfair die

You are given a 6-sided die.The die is biased, so it is more likely to produce some results than others, but you don't know the exact odds. It is however given that the odds are the same for each roll,...
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### Three Dice minimum value

You shall form three dice, placing 18 distinct integers on the faces of three cubes. Your goal is to be able to obtain all the integers between 1 and 216, inclusive, as the sum of the integers on the ...
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### A Short Dice Puzzle

There are $2$ fair dice: An $11$-sided die, valued from $-5$ to $5$, A $41$-sided die, valued from $-20$ to $20$. You pick one, and I'll take the other one. We roll the dice, and whoever rolls the ...
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### Chain Puzzle: Tabletop Games #05 - It's Yahtzee, Jim, but not as we know it

This is the fifth Chain Puzzle in the Tabletop Games series, in which all puzzles are themed around board games, card games, tile games, and the like. The answer to this puzzle is a thematic word or ...
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### How many rolls on average to get N dice to all show the same value?

So for example: if $N = 3$ I roll and get -> $1,2,3$ I re-roll the $2$ and $3$ -> $1,4,4$ I re-roll the $1$ -> $2,4,4$ I re-roll the $2$ -> $4,4,4$ In this case, it ...
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### How to get a uniform 1 out of 7 chance, using only a coin and a 6-sided die [closed]

There are seven days in a week. Let's say you want to pick one at random, such that each day has an equal chance of being picked as each other day. In other words, the goal is to obtain a uniform ...
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### Generating Roman numerals with dice

This puzzle is closely based on this one: Generating numbers with cubes Now we want to generate Roman numerals by placing up to three 6-sided dice side by side. We are allowed to write multiple ...
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### Rolling Dice puzzle

The answer is a six-letter word. Here's an attempt at a text version, but you're really better off looking at the picture. Rolling Dice To get a full tour, you must be a man of many faces and ...
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### Two Dice = Nine Cards

I have nine cards, each labeled with a real number (possibly negative or fractional) in a special way. When I randomly pick two of these cards without replacement and add their labels together, the ...
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### Loaded Dice with Uniform Sum

A loaded die looks like a regular six-sided die, but instead of each number 1-6 being equally likely, some numbers may be more or less likely than others. Is it possible to have two loaded dice so ...
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### The Sphinx's Dice

"I'm bored of riddles, I'm going to ask you a question about dice instead", says the Sphinx. "Now, I've got this set of Efron Dice here. There's four dice, and they have these values on their faces:" ...
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### Date Dice Dilemma

My cousin just learned about numbers and dates in school and I would like to give him a little present: A set of (regular, six-sided) dice with which he can display the current date if he turns them ...
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### Crazy Dice - Unusual Dice with Usual Sum

Can you design two six-sided dice (different from the standard ones), where each face has a nonzero number of dots, so that the probability distribution of their total is the same as for two standard ...
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### How to simulate one die with three dice?

I do not know if there is a correct solution to this puzzle, but I can't stop thinking about it, so here goes: is it possible to accurately simulate a single, fair, six-sided die using three custom ...
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### Find the optimal strategy for this dice game

You are playing a game: You have $6$ standard 6-sided dice that are all rolled at the start of the game. The sum of dice values is your score. Each turn, you can take any (or none) dice and reroll ...
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### Recipe for diced shark fin

How may the following shark-fin-shaped Goal distribution be cooked up with 44 ingredients in a roll recipe R based on the roll sum of two 6-sided dice with non-standard allotments of dots? Goal: ...
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### A Lazy Way to Make Two Dice from Many

Recently, I stumbled upon the puzzle: Make Two Dice out of Three, which asks for a solution to a case of the following question: Suppose I roll $n$ indistinguishable dice. From purely the $n$ ...
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### Putting the pips on a d6

Using a blank cube and a bunch of circular stickers, an average person constructs a d6. A d6 is also known as a six-sided die, or sometimes, a dice. For the purpose of this puzzle, the average person ...
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### What is the 'best' way to tilt through a D20?

A regular D20 die (icosahedron) as depicted above has the number 1 to 20 arranged on its faces such that the sum of opposing faces is 21. Each triangular face touches three other faces directly by a ...
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### A first choice die

You and your friend have met to play a game of dice. This is a very simple game where both players roll a die at the same time, and the player with the lower score pays a coin to the other player. In ...
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### Cheating aplenty at Build-a-Die 2017

Each year, 9 affluent acquaintances meet to compete in a unique game of chance: each creates a fair six-sided die and rolls against each opponent 100 times -- the player with the most total wins in ...
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### 2 of 4 cards for 1 die

These are equivalent to using using a deck of 4 cards to emulate a 6-sided die. A. Assign integer values to   a, b, c, d   so that the 6 possible &...
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### Four dice puzzle: What's the best throw?

This continues Damiano's puzzle "Four dice puzzle: 2,2,4,5" Damiano keeps throwing his four dice. After a lot of throwing and thinking and working, he has determined for every throw $a,b,c,d$ of his ...
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### Rolling dice in a box

A box containing identical dice that were arranged to have same orientation has a missing die at the center (see figure). The dice moves to the free space for they can change orientation and ...
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### Building the ideal Scrabble dice set

This is a genuine question and I don't know the "correct" answer to this. I'm also of two minds if it is really on-topic on the site. Finding the answer is surely "a puzzle" (if ...
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### Four dice puzzle: 2,2,4,5

Damiano has thrown four dice and the numbers 2, 2, 4, 5 showed up on top. Damiano asks himself: What is the smallest positive integer that cannot be generated with these four numbers according to ...
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### Stickered Dice Game

You and your friend play a game. To begin with, each of you has an unlabelled die. On the table laid in front of you are 12 stickers labelled 1-12. The first player chooses one of the stickers and ...
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### Fair d5 with as few faces as possible

Challenge: design a polyhedral die that will always give one of five outcomes, each with equal probability. While achieving that, minimize the total number of faces of the die. Think of the die as ...
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### A Pair of Odd (but Still Balanced) Dice

I have a pair of odd six-sided dice at my house... I don't quite remember where I got them. Each die has only one number on each side, and each of these numbers is a positive number. Each pair of ...
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### West face East face

The 1 x 1 x 2 die with new configuration was created from 2 dice (with opposite faces totalling 7) that were glued together on one of their faces. If this long die is rolled twice, the sum of non-...
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