Directions : A word and number arrangement machine when given an input line of…

Concept

An input-output arrangement machine transforms a line of words and numbers through a fixed sequence of steps. Words and numbers evolve under separate, parallel rules that stay constant from step to step. To solve a new input, first decode each rule from the worked illustration, then apply the identical rules, in the identical order, to the new input.

Decoding the number rule

Working from the illustration, each Step V number is reached from the original number through four fixed moves.

1. Step I: for an original 4-digit number with digits A, B, C, D, Step I number = concat(A2, B+1, C+1, D-1). For example 5247 (A=5, B=2, C=4, D=7) gives 25, then 3, then 5, then 6, i.e. 25356, matching Step I in the illustration; the same construction reproduces all five Step I numbers (3827 -> 9936, 8425 -> 64534, 2843 -> 4952, 9832 -> 81941).

2. Step II only re-orders the Step I number's own digits (same digits, e.g. 25356 becomes 26355) - this does not change the sum of the even-valued digits or the sum of the odd-valued digits, so that sum can be read straight off the Step I number.

3. Step III: split the digits into E, the sum of the even-valued digits, and O, the sum of the odd-valued digits - e.g. for 25356, E = 8 (2+6) and O = 13 (5+3+5).

4. Step IV = O2 - E, e.g. 132 - 8 = 161.

5. Step V = Step IV + 3 when Step IV is odd, or Step IV + 4 when Step IV is even, e.g. 161 (odd) + 3 = 164.

Applying the rule to the given input

Sum of the five Step V numbers = 132 + 50 + 248 + 134 + 350 = 914.

Cross-check

Running the identical four moves on the illustration's own numbers reproduces every printed value exactly: 5247 -> 25356 -> (E=8, O=13) -> 161 -> 164; 3827 -> 9936 -> (E=6, O=21) -> 435 -> 438; 8425 -> 64534 -> (E=14, O=8) -> 50 -> 54; 2843 -> 4952 -> (E=6, O=14) -> 190 -> 194; 9832 -> 81941 -> (E=12, O=11) -> 109 -> 112 - confirming the decoded rule.

So the sum of the numbers in Step V for the new input is 914.