##
   ##
  ###
+ ###
 ####

In the above addition all the digits were replaced by "#".
Please restore the original numbers, given that in each column the digits form a valid arithmetic progression.

• Based upon a puzzle by Polish mathematician Marek Penshko.

(In reply to Other solutions by tomarken)

If we're allowed to use d = 0 for an arithmetic progression then I found these additional solutions. 

34 + 33 + 632 + 331 = 1030
34 + 33 + 732 + 531 = 1330
34 + 33 + 832 + 731 = 1630
34 + 33 + 932 + 931 = 1930

68 + 66 + 664 + 462 = 1260
68 + 66 + 764 + 662 = 1560
68 + 66 + 864 + 862 = 1860

80 + 60 + 640 + 420 = 1200
80 + 60 + 740 + 620 = 1500
80 + 60 + 840 + 820 = 1800

40 + 30 + 620 + 310 = 1000
40 + 30 + 720 + 510 = 1300
40 + 30 + 820 + 710 = 1600
40 + 30 + 920 + 910 = 1900

I think that may be it, but I worked out the logic by hand so I'm not sure if I exhausted every possibility. 

Edited to add two I left out:

28 + 36 + 944 + 952 = 1960

74 + 63 + 852 + 841 = 1830

I'm pretty sure that these, combined with those in my prior post, are all the possible solutions. 

Edited on May 19, 2014, 3:25 pm

Posted by tomarken on 2014-05-19 14:02:46