"Get the proof of his age, Joe".

Insert the 4 bold words into alphametic equations:

(I) W1+W2=W3
(II) W1-W2=W4

in such an order that there will be a unique set of non-zero digits (each distinct letter representing a distinct digit) fitting as a solution.

Which digits represent the nine letters a,e, ...t?

REM: I've built this puzzle bottom-up, so there might be solutions different from mine - unlikely, but always welcome.

(In reply to No real spoiler by brianjn)

What I had said in the queue was that one way of making the solution unique would have been to limit the digits to the range 0 through 8, rather than specifying non-zero digits.

The program below sought solutions leaving out any one of the ten digits.

DECLARE SUB permute (a$)
CLS
'     gethisajo
a$ = "1234567890": h$ = a$
DO
  IF MID$(a$, 1, 1) > "0" AND MID$(a$, 4, 1) > "0" AND MID$(a$, 7, 1) > "0" AND MID$(a$, 8, 1) > "0" THEN
    getv = VAL(LEFT$(a$, 3))
    hisv = VAL(MID$(a$, 4, 3))
    agev = VAL(MID$(a$, 7, 1) + MID$(a$, 1, 2))
    joev = VAL(MID$(a$, 8, 2) + MID$(a$, 2, 1))
    IF getv + hisv = agev AND ABS(getv - hisv) = joev THEN
      PRINT "g+h=a", getv; hisv; agev; joev, RIGHT$(a$, 1)
    END IF
    IF getv + hisv = joev AND ABS(getv - hisv) = agev THEN
      PRINT "g+h=j", getv; hisv; joev; agev, RIGHT$(a$, 1)
    END IF
    IF getv + agev = hisv AND ABS(getv - agev) = joev THEN
      PRINT "g+a=h", getv; agev; hisv; joev, RIGHT$(a$, 1)
    END IF
    IF getv + agev = joev AND ABS(getv - agev) = hisv THEN
      PRINT "g+a=j", getv; agev; joev; hisv, RIGHT$(a$, 1)
    END IF
    IF getv + joev = hisv AND ABS(getv - joev) = agev THEN
      PRINT "g+j=h", getv; joev; hisv; agev, RIGHT$(a$, 1)
    END IF
    IF getv + joev = agev AND ABS(getv - joev) = hisv THEN
      PRINT "g+j=a", getv; joev; agev; hisv, RIGHT$(a$, 1)
    END IF
    IF hisv + agev = getv AND ABS(hisv - agev) = joev THEN
      PRINT "h+a=g", hisv; agev; getv; joev, RIGHT$(a$, 1)
    END IF
    IF hisv + agev = joev AND ABS(hisv - agev) = getv THEN
      PRINT "h+a=j", hisv; agev; joev; getv, RIGHT$(a$, 1)
    END IF
    IF hisv + joev = getv AND ABS(hisv - joev) = agev THEN
      PRINT "h+j=g", hisv; joev; getv; agev, RIGHT$(a$, 1)
    END IF
    IF hisv + joev = agev AND ABS(hisv - joev) = getv THEN
      PRINT "h+j=a", hisv; joev; agev; getv, RIGHT$(a$, 1)
    END IF
    IF agev + joev = getv AND ABS(agev - joev) = hisv THEN
      PRINT "a+j=g", agev; joev; getv; hisv, RIGHT$(a$, 1)
    END IF
    IF agev + joev = hisv AND ABS(agev - joev) = getv THEN
      PRINT "a+j=h", agev; joev; hisv; getv, RIGHT$(a$, 1)
    END IF
  END IF
  permute a$
LOOP UNTIL a$ = h$

The results are tabulated below. The first row for zero being unused is:

g+j=h          524  372  896  152         0

which is translated as

524 + 372 = 896
get + joe = his
get - joe = age
524 - 372 = 152

The other three are deciphered in the same way:

h+j=g          456  283  739  173         0
h+a=g          568  174  742  394         0
h+j=g          568  374  942  194         0

g stands for get
h stands for his
a stands for age
j stands for joe

If the unused digit had been 9 the answer would have been unique:

a+j=h          652  132  784  520         9

652   132   784 
age + joe = his
age - joe = get
652   132   520

the            same order   remaining   digit 
addition       as at left   word value not used

h+a=j          380  216  596  164         7
h+a=j          540  321  861  219         7
h+a=j          570  324  894  246         1
a+j=h          627  357  984  270         1
a+j=h          429  139  568  290         7
h+a=j          460  132  592  328         7
a+j=h          534  194  728  340         6
a+j=h          652  132  784  520         9
g+j=h          524  372  896  152         0
h+a=j          690  153  843  537         2
a+j=h          754  214  968  540         3
g+a=h          548  154  702  394         6
g+a=j          610  261  871  349         5
h+j=a          820  147  967  673         5
h+j=g          456  283  739  173         0
h+a=g          568  174  742  394         0
g+a=j          760  176  936  584         2
a+j=g          576  186  762  390         4
a+j=g          579  219  798  360         4
g+j=a          860  126  986  734         5
h+j=g          568  374  942  194         0

Posted by Charlie on 2013-03-26 22:49:13