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Task/Babbage-problem/J/babbage-problem-1.j

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+   square=: ^&2
+   modulo1e6=: 1000000&|
+   trythese=: i. 1000000                   NB. first million nonnegative integers
+   which=: I.                              NB. position of true values
+   which 269696=modulo1e6 square trythese  NB. right to left <-
+25264 99736 150264 224736 275264 349736 400264 474736 525264 599736 650264 724736 775264 849736 900264 974736

Task/Babbage-problem/J/babbage-problem-2.j

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+NB. In the interactive environment.
+NB. First here, Mr Babbage, we'll make the computer's words more meaningful to an english speaker.
+
+NB. The first is the "head" of a list, written with these inviting open arms that embrace one small dot :
+   first=: {.
+
+NB. The small i. notation denotes "all integers up to 100000". You've already found a solution in that range.
+   n=: i. 100000
+
+NB. This is how we write squaring.
+   squareof=: *:
+
+NB. In our notation, a dyad is a word that takes an x value on the left and an y value on the right.
+   ends=: dyad : ' x = 1000000 | y '
+
+NB. This dyad selects values from the list x, as marked by the list y
+   where=: dyad : ' y # x '
+
+NB. Now that we defined our words, we can ask our question with them :
+   first n where 269696 ends squareof n
+25264
+
+NB. With a bit of habit, you won't need to define words in english anymore.
+NB. The following easily relates word for word to the sentence we've written :
+   {. (i.100000) #~ 269696 = 1000000 | *: i.100000
+25264
+
+NB. Like all mathematical notations, in J you see patterns that suggest simplification :
+   {. I. 269696 = 1000000 | *: i.100000
+25264