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| '''Sicherman dice''' {{IPAc-en|ˈ|s|ɪ|k|ər|m|ən}} are the only pair of 6-sided [[dice]] that are not [[normal dice]], bear only [[positive integers]], and have the same [[probability distribution]] for the [[sum]] as normal dice.
| | Hello and welcome. My name is Numbers Wunder. Puerto Rico is exactly where he's usually been residing but she requirements to move because of her family. Doing ceramics is what my family members and I enjoy. Hiring is his profession.<br><br>Review my website ... std home test ([http://www.buzzbit.net/blog/333162 see here]) |
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| The faces on the dice are numbered 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8.
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| ==Mathematics==
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| '''Crazy dice''' is a standard [[mathematics|mathematical]] problem or puzzle in elementary [[combinatorics]], involving a re-labeling of the faces of a pair of six-sided [[dice]] to reproduce the same frequency of [[sums]] as the standard labeling.
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| It is a standard exercise in elementary combinatorics to calculate the number of ways of rolling any given value with a pair of fair six-sided dice (by taking the sum of the two rolls). The table shows the number of such ways of rolling a given value <math>n</math>:
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| {| class="wikitable"
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| |-
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| | align=right | ''n''
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| | align=right | 2
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| | align=right | 3
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| | align=right | 4
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| | align=right | 5
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| | align=right | 6
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| | align=right | 7
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| | align=right | 8
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| | align=right | 9
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| | align=right | 10
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| | align=right | 11
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| | align=right | 12
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| |-
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| | align=right | # of ways
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| | align=right | 1
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| | align=right | 2
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| | align=right | 3
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| | align=right | 4
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| | align=right | 5
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| | align=right | 6
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| | align=right | 5
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| | align=right | 4
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| | align=right | 3
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| | align=right | 2
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| | align=right | 1
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| |}
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| A question arises whether there are other ways of re-labeling the faces of the dice with [[positive integer]]s that generate these sums with the same frequencies. The surprising answer to this question is that there does indeed exist such a way. These are the Sicherman dice.
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| The table below lists all possible totals of dice rolls with standard dice and Sicherman dice. One Sicherman die is coloured for clarity: '''<span style="color:green;">1</span>–<span style="color:red;">2</span>–<span style="color:blue;">''2''</span>–<span style="color:red;">3</span>–<span style="color:blue;">''3''</span>–<span style="color:green;">4</span>''', and the other is all black, 1–3–4–5–6–8.
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| {| class="wikitable"
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| |-
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| | align=centre |
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| | align=centre | 2
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| | align=centre | 3
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| | align=centre | 4
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| | align=centre| 5
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| | align=centre| 6
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| | align=centre| 7
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| | align=centre| 8
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| | align=centre| 9
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| | align=centre| 10
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| | align=centre| 11
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| | align=centre| 12
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| |-
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| |Standard dice
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| |1+1
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| |1+2<br />2+1
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| |1+3<br />2+2<br />3+1
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| |1+4<br />2+3<br />3+2<br />4+1
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| |1+5<br />2+4<br />3+3<br />4+2<br />5+1
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| |1+6<br />2+5<br />3+4<br />4+3<br />5+2<br />6+1
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| |2+6<br />3+5<br />4+4<br />5+3<br />6+2
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| |3+6<br />4+5<br />5+4<br />6+3
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| |4+6<br />5+5<br />6+4
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| |5+6<br />6+5
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| |6+6
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| |-
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| |Sicherman dice
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| | '''<span style="color:green;">1</span>'''+1
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| | '''<span style="color:red;">2</span>'''+1<br />'''<span style="color:blue;">''2''</span>'''+1
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| | '''<span style="color:red;">3</span>'''+1<br />'''<span style="color:blue;">''3''</span>'''+1<br />'''<span style="color:green;">1</span>'''+3
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| | '''<span style="color:green;">1</span>'''+4<br />'''<span style="color:red;">2</span>'''+3<br />'''<span style="color:blue;">''2''</span>'''+3<br />'''<span style="color:green;">4</span>'''+1
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| | '''<span style="color:green;">1</span>'''+5<br />'''<span style="color:red;">2</span>'''+4<br />'''<span style="color:blue;">''2''</span>'''+4<br />'''<span style="color:red;">3</span>'''+3<br />'''<span style="color:blue;">''3''</span>'''+3
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| | '''<span style="color:green;">1</span>'''+6<br />'''<span style="color:red;">2</span>'''+5<br />'''<span style="color:blue;">''2''</span>'''+5<br />'''<span style="color:red;">3</span>'''+4<br />'''<span style="color:blue;">''3''</span>'''+4<br />'''<span style="color:green;">4</span>'''+3
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| | '''<span style="color:red;">2</span>'''+6<br />'''<span style="color:blue;">''2''</span>'''+6<br />'''<span style="color:red;">3</span>'''+5<br />'''<span style="color:blue;">''3''</span>'''+5<br />'''<span style="color:green;">4</span>'''+4
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| | '''<span style="color:green;">1</span>'''+8<br />'''<span style="color:red;">3</span>'''+6<br />'''<span style="color:blue;">''3''</span>'''+6<br />'''<span style="color:green;">4</span>'''+5
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| | '''<span style="color:red;">2</span>'''+8<br />'''<span style="color:blue;">''2''</span>'''+8<br />'''<span style="color:green;">4</span>'''+6<br />
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| | '''<span style="color:red;">3</span>'''+8<br />'''<span style="color:blue;">''3''</span>'''+8
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| | '''<span style="color:green;">4</span>'''+8
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| |}
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| ==History==
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| These dice were discovered by George Sicherman of [[Buffalo, New York]] and were originally reported by [[Martin Gardner]] in a 1978 article in ''[[Scientific American]]''.
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| The numbers can be arranged so that all pairs of numbers on opposing sides sum to equal numbers, 5 for the first and 9 for the second.
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| Later, in a letter to Sicherman, Gardner mentioned that a magician he knew had anticipated Sicherman's discovery. For generalizations of the Sicherman dice to more than two dice and noncubical dice, see Broline (1979), Gallian and Rusin (1979), Brunson and Swift (1997/1998), and Fowler and Swift (1999).
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| ==Mathematical justification==
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| Let a ''canonical'' ''n''-sided die be an [[polyhedron|''n''-hedron]] whose faces are marked with the integers [1,n] such that the probability of throwing each number is 1/''n''. Consider the canonical cubical (six-sided) die. The [[generating function]] for the throws of such a die is <math>x + x^2 + x^3 + x^4 + x^5 + x^6</math>. The product of this polynomial with itself yields the generating function for the throws of a pair of dice: <math>x^2 + 2 x^3 + 3 x^4 + 4 x^5 + 5 x^6 + 6 x^7 + 5 x^8 + 4 x^9 + 3 x^{10} + 2 x^{11} +x^{12}</math>. From the theory of [[Root of unity#Cyclotomic polynomials|cyclotomic polynomials]], we know that
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| :<math>x^n - 1 = \prod_{d\,\mid\,n}^n \Phi_d(x).\;</math>
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| where ''d'' ranges over the [[divisor]]s of ''n'' and <math>\Phi_d(x)\,</math> is the ''d''-th cyclotomic polynomial. We note also that
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| :<math>\frac{x^n -1}{x-1} = \sum_{i=0}^{n-1} x^i = 1 + x + \cdots + x^{n-1}</math>.
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| We therefore derive the generating function of a single ''n''-sided canonical die as being
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| :<math>x + x^2 + \cdots + x^n = \frac{x}{x-1} \prod_{d\,\mid\,n}^n \Phi_d(x)</math>
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| <math>\Phi_1(x) = x - 1\,</math> and is canceled. Thus the [[factorization]] of the generating function of a six-sided canonical die is
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| :<math>x\,\Phi_2(x)\,\Phi_3(x)\,\Phi_6(x) = x\;(x+1)\;(x^2 + x + 1)\;(x^2 - x +1)</math>
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| The generating function for the throws of two dice is the product of two copies of each of these factors. How can we partition them to form two legal dice whose spots are not arranged traditionally? Here ''legal'' means that the coefficients are non-negative and sum to six, so that each die has six sides and every face has at least one spot. (That is, the generating function of each die must be a polynomial p(x) with positive coefficients, and with p(0) = 0 and p(1) = 6.)
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| Only one such partition exists:
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| :<math>x\;(x + 1)\;(x^2 + x + 1) = x + 2x^2 + 2x^3 + x^4</math>
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| and
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| :<math>x\;(x + 1)\;(x^2 + x + 1)\;(x^2 - x + 1)^2 = x + x^3 + x^4 + x^5 + x^6 + x^8</math>
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| This gives us the distribution of spots on the faces of a pair of Sicherman dice as being {1,2,2,3,3,4} and {1,3,4,5,6,8}, as above.
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| This technique can be extended for dice with an arbitrary number of sides.
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| ==References==
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| *{{citation
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| | doi = 10.2307/2689786
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| | last = Broline
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| | first = D.
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| | title = Renumbering of the faces of dice
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| | journal = Mathematics Magazine
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| | volume = 52
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| | issue = 5
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| | year = 1979
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| | pages = 312–315
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| | jstor = 2689786
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| | publisher = Mathematics Magazine, Vol. 52, No. 5}}
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| *{{citation
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| | last1 = Brunson
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| | first1 = B. W.
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| | last2 = Swift
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| | first2 = Randall J.
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| | title = Equally likely sums
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| | journal = Mathematical Spectrum
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| | volume = 30
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| | issue = 2
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| | year = 1997/8
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| | pages = 34–36}}
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| *{{citation
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| | last1 = Fowler
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| | first1 = Brian C.
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| | last2 = Swift
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| | first2 = Randall J.
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| | title = Relabeling dice
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| | journal = College Mathematics Journal
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| | volume = 30
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| | issue = 3
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| | year = 1999
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| | pages = 204–208
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| | jstor = 2687599
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| | doi = 10.2307/2687599
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| | publisher = The College Mathematics Journal, Vol. 30, No. 3}}
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| *{{citation
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| | last1 = Gallian
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| | first1 = J. A.
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| | last2 = Rusin
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| | first2 = D. J.
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| | title = Cyclotomic polynomials and nonstandard dice
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | volume = 27
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| | year = 1979
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| | pages = 245–259
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| | doi = 10.1016/0012-365X(79)90161-4
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| | mr = 0541471
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| | issue = 3}}
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| *{{citation
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| | last = Gardner
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| | first = Martin
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| | authorlink = Martin Gardner
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| | title = Mathematical Games
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| | journal = [[Scientific American]]
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| | year = 1978
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| | volume = 238
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| | issue = 2
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| | pages = 19–32
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| | doi = 10.1038/scientificamerican0278-19}}
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| == External links ==
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| *[http://www.grand-illusions.com/acatalog/Sicherman_Dice.html Grand Illusion's Informational Page]
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| *[http://mathworld.wolfram.com/SichermanDice.html Mathworld's Information Page]
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| {{PlanetMath attribution|id=6738|title=Crazy dice}}
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| [[Category:Dice]]
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| [[Category:Combinatorics]]
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Hello and welcome. My name is Numbers Wunder. Puerto Rico is exactly where he's usually been residing but she requirements to move because of her family. Doing ceramics is what my family members and I enjoy. Hiring is his profession.
Review my website ... std home test (see here)