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 Upon Reflection (Posted on 2013-11-22)
The Alphametics:
WED + TAB + PIN = X j k Y
and
DEW + BAT + NIP = Y k j X
..... when considered concurrently have 864 solutions.

In the totals X=1 and Y=2 or X=2 and Y=1, and j and k are distinct integers represented in the set of variables.

Why can X and Y never equate to W, T, P, D, B or N? The digits 0-9 are all available but there are no leading zeroes.

 Submitted by brianjn No Rating Solution: (Hide) This has 864 permutation solutions for which there are 2 permuted solutions. When written as:``` W E D and D E W T A B B A T P I N N I P X j k Y Y k j X ``` it quickly becomes obvious that X can only be 1 or 2, and so by reflection then Y is whatever X is not. If X = 2 then W + T + P = 22 to take into account that upon reflection the thousands and units digits must be 2. Similarly if Y = 1 then D + B + N = 11. For a total of 22 each of the digit addends must be at least 3 which precludes X and Y in this case. 22 will be the sum of the digits of either 9,8 & 5 or 9,7 & 6. The sum E + A + I cannot equal 1, and once the sets of 22 and 11 are satisfied these remaining digits must total 11. Compiling these place value totals, 22,11,11 and 11,11,22 renders:``` X j k Y = 2321 or 1232 and Y k j X = 1232 or 2321 ``` If 11 has 1 within is summation it is formed from the digits: 1,2,8; 1,3,7 or 1,4,6. Paired with the make-up of the 22 totals, and the remaining digits(R), the following table offers:```(11) (R) (22) (11) (R) (22) (11) (R) (22) 1 0 9 1 0 9 1 0 9 6 2 8 7 2 8 8 3 7 4 3 5 3 4 5 2 4 6 7 6 5``` In all cases remaining digits(R) total 12! Dismissing the "1s" in the (11) columns and swapping the underlined digits (1 for 2) in each case forms the balance 11,11,22. 1 cannot appear as any of the digital addends for 11. While it may be noted that 2 can occur as an addend for 11 the problem did state: "Why can X and Y never equate to ... ".

 Subject Author Date re(2): The Essence - re:official solution Ady TZIDON 2013-12-04 00:08:01 re: The Essence - re:official solution brianjn 2013-11-29 19:39:26 The Essence - re:official solution Ady TZIDON 2013-11-28 03:36:02 The Style Ady TZIDON 2013-11-28 03:24:24 re: Hmmph. Ady TZIDON 2013-11-28 03:21:41 Hmmph. brianjn 2013-11-26 21:36:33 re(3): Disappointed! - I AM HAPPY- now resolved Ady TZIDON 2013-11-25 05:10:50 re(2): Disappointed! - I AM brianjn 2013-11-25 03:25:28 re: Disappointed! - I am not Ady TZIDON 2013-11-25 02:45:58 Disappointed! brianjn 2013-11-24 04:34:25 My way Ady TZIDON 2013-11-23 15:21:18 re: Is this it? ....not quite Ady TZIDON 2013-11-23 13:48:40 Is this it? Charlie 2013-11-22 12:56:19

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