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Upon Reflection (Posted on 2013-11-22) Difficulty: 3 of 5
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    
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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 ... ".

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre(2): The Essence - re:official solutionAdy TZIDON2013-12-04 00:08:01
re: The Essence - re:official solutionbrianjn2013-11-29 19:39:26
Some Thoughts The Essence - re:official solutionAdy TZIDON2013-11-28 03:36:02
Some ThoughtsThe StyleAdy TZIDON2013-11-28 03:24:24
Some Thoughtsre: Hmmph.Ady TZIDON2013-11-28 03:21:41
Hmmph.brianjn2013-11-26 21:36:33
Some Thoughtsre(3): Disappointed! - I AM HAPPY- now resolvedAdy TZIDON2013-11-25 05:10:50
re(2): Disappointed! - I AMbrianjn2013-11-25 03:25:28
Some Thoughtsre: Disappointed! - I am notAdy TZIDON2013-11-25 02:45:58
Disappointed!brianjn2013-11-24 04:34:25
SolutionMy wayAdy TZIDON2013-11-23 15:21:18
Hints/Tipsre: Is this it? ....not quiteAdy TZIDON2013-11-23 13:48:40
Some ThoughtsIs this it?Charlie2013-11-22 12:56:19
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