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Using Mathematica to solve Smullyan "Scheherazade" puzzle?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg72281] Using Mathematica to solve Smullyan "Scheherazade" puzzle?
*From*: "Kelly Jones" <kelly.terry.jones at gmail.com>
*Date*: Mon, 18 Dec 2006 06:55:43 -0500 (EST)
Can Mathematica be used to solve this puzzle from Raymond Smullyan's
"The Riddle of Scheherazade"? (isbn://0156006065/page/74)
Smullyan defines the following function. It's really a string-based
function, not a numerical one, even though it only uses the characters 0-9.
In the below, x refers to any string of digits, and "xy" on the left
side means the concatenation of the strings x and y, not
multiplication. Not all functions are in proper Mathematica form.
f["1x2"] = x;
(* eg: f["13542"] = "354" *)
f["3x"] = StringJoin[f[x], f[x]];
(* eg: f["315432"] = StringJoin[f["15432"], f["15432"]] =
StringJoin["543", "543"] = "543543" *)
f["4x"] = StringReverse[f[x]];
(* eg: f["4172962"] = StringReverse[f["172962"]] =
StringReverse["7296"] = "6927" *)
f["5x"] = If[StringLength[f[x]]>1, StringDrop[f[x],1], f[x]]]
(* eg: f["513472"] =
If[StringLength[f["13472"]]>1, StringDrop[f["13472"],1], f["13472"]]] =
If[StringLength["347"]>1, StringDrop["347",1], "347"] =
If[True, "47", "347"] = "47"
*)
f["6x"] = StringJoin["1", f[x]]
(* eg: f["615832"] = StringJoin["1", f["15832"]] = StringJoin["1", "583"] =
"1583"
*)
f["7x"] = StringJoin["2", f[x]]
(* eg: f["715832"] = StringJoin["2", f["15832"]] = StringJoin["2", "583"] =
"2583"
*)
The rules are, of course, "recursive". Example:
f["33x"] = f["3x"]f["3x"] = f[x]f[x]f[x]f[x]
The function isn't necessarily defined for every string.
Smullyan now asks us to find values such that:
f[x] == x;
f[x] == StringJoin[x,x];
f[x] == StringReverse[x]; (* StringLength[x] <= 12 if possible *)
f[x] == StringDrop[x, -1];
f[f[x]] == x && f[x] != x;
f[f[x]] == StringReverse[x];
f[f[x]] == StringJoin[StringTake[x,-1], StringTake[x,{2,-2}], StringTake[x,1]]
(* right side is x with first and last digits reversed *)
StringTake[f[StringReverse[f[x]]], StringLength[f[StringReverse[f[x]]]]/2] ==
StringTake[f[StringReverse[f[x]]], -StringLength[f[StringReverse[f[x]]]]/2]
&&
f[StringTake[f[StringReverse[f[x]]], StringLength[f[StringReverse[f[x]]]]/2] ==
x;
(* in other words, f[StringReverse[f[x]]] is of the form "zz", and f["z"] == x)
Of course, the answers are in the back of the book, so my general
question is: can Mathematica be used to solve puzzles of this sort?
Enumerating and testing all possible strings seems tedious (especially
given the length of some of the answers). Is there a clever
shortcut/methodology here?
--
We're just a Bunch Of Regular Guys, a collective group that's trying
to understand and assimilate technology. We feel that resistance to
new ideas and technology is unwise and ultimately futile.
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