Votes for this machine would be recorded on 1/2" paper tapes which would be factory-punched with sprocket holes every 1/4" down one side. Each tape would be marked with a unique design (preferably identifying the race, candidate, district, ward, and machine) which repeated every 1/4" or 1/2" (lined up with the sprocket holes). There would be one tape for each candidate, plus one for each race for a write-in candidate, and one for each race for "no candidate selected". An additional tape would be used in any "vote for two" race to indicate that only one candidate was selected [two extras in vote-for-three or vote-for-four races].
Initially, the tapes would have no punches in them other than the sprocket holes. Before the election, each tape would be marked at the end to indicate the race, precinct, ward, and machine for which the tape would be used as well as the candidate the tape represents. It would be further signed and sealed by representatives of all interested parties. [One set of seals could be applied by a surely-trusted representative of each party who need not be present on-site and another set applied by on-site representatives].
To the extent possible, everything in the machine would be transparent except the enclosures holding the tape spools. Each tape would come out of an opaque enclosure, through a visible mechanism, and proceed into another opaque enclosure. After the voter selects candidates using pull-levers and pulls the "finish voting" lever, the visible mechanism would, in clear view of the voter, punch the appropriate tape for every selected office (or two tapes for vote-for-two offices) and advance the appropriate tapes by 1/4" (or 1/2", depending upon the design size).
If there are four inches of tape between the punching mechanism and the opaque takeup-spool enclosure, it would be necessary to prep the machine by punching each tape ten times after loading, to ensure that every voter saw punched tape from the puncher to the takeup spool; these ten punches would be figured in the tally below.
Once voting is complete, the tail end of each tape would be marked and sealed by appropriate party representatives and the tapes submitted for counting. Counting could be accomplished very quickly and easily, since each tape would simply have a uniform bunch of holes on it. Subtract ten from the number of holes to get the number of votes for the candidate represented by the tape.
The use of a "no candidate selected" tape would protect other tapes from alteration; if 2,500 people voted and there were six presidential candidates (plus "write-in" and "no-candidate-selected") then there should be exactly 2,580 non-sprocket holes in all the presidential tapes. Assuming the tape is designed so that patching holes is impossible, and assuming that the number of voters is logged immediately upon the close of the election, it would be difficult for anyone to perform fraud upon the tapes without detection.
Anyone like that as an approach? The mechanics might be a little bit complicated, but hopefully not too bad. I think it would give voters a unique ability to actually see their vote permanently and indelibly recorded, something which is lacking in other voting systems.