For definiteness, I take Henry's question to be, "Is there a puzzle with the original Foxtrot structure, that has a unique solution in every radix starting with some threshold?" I found these five: # dif syms symbol pattern divisor quotient radix 5 ABACBBDABAZ 1 2 1 r-1 5..100 6 AABCDBEAABC 2 2 1 r-1 7..100 6 ABCDBCEABCD 2 4 1 r-3 17..100 (and 15) 7 ABCDBEFABAC 2 4 1 r-1 10..100 (and 7) 7 ABCDEEFABAC 3 5 1 r-1 12..100 (and 9) Reminder: the symbol pattern is the symbols for digit positions p1 through p11, in order. Symbol Z=0 is implicitly present, even if it does not occur in p1 through p11. The original Foxtrot structure is: p3 p4 ------------ p1 p2 ) p5 p6 p7 p8 p9 ------- p10 p11 p7 p10 p11 p7 ---------- Z The first symbol pattern above, ABACBBDABAZ, makes: A C ---------- A B ) B B D A B ---- A Z D A Z D ------- Z And that has the unique solution, for radix 5 through 100: 1 r-1 ------------ 1 2 ) 2 2 r-2 1 2 ---- 1 0 r-2 1 0 r-2 --------- 0 A short form to speccify this puzzle and solution is: (2 2 r-2) / (1 2) = (1 r-1) (radix r = 5 to at least 100) Existence of a unique solution for each symbol pattern was checked through radix=100. For the first two and last two, I think I proved a unique solution for all r; but a proof eludes me for the third one. That worries me, because another pattern, ABACDEFEDGC, has a unique solution (divisor = 2 3, quotient = 2 r-2) for radix 13..52; but for radix 53 it has a second solution (divisor = 6 34, quotient = 6 47)! — Mike
On Jan 18, 2019, at 2:55 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Mike:
Perhaps this is already known, but what about long division puzzles which have unique solutions *for each radix*, but can be (uniquely) solved for multiple radices (radishes?) ?
BTW, one of the early claims for the Prolog programming language was its ability to "trivially" solve digit arithmetic puzzles; Prolog automatically performed "hypothesize & test" for you, with builtin backtracking when an hypothesis failed.
On Jan 13, 2019, at 5:39 PM, rcs@xmission.com wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
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