Errata: Challenge #21


To solve puzzle #21 with the advertised 5 additional ramps, the
starting position has to be different; with the bottom ramp and
the one immediately above it pointed left instead of right. Or
alternatively use the starting position and 7 ramps.


I disagree - I solved it with the given layout and 5 more ramps.

Though, grammatically, the Objective should say “Use 15 or fewer balls” rather than “Use 15 or less balls”


I suspect you violate the “long drop” rule.


Just so we’re 100% clear, we’re talking about taking this layout (number of blue balls may vary):

And adding 5 ramps to it.

My solution is which matches the solution in the book. No long drops.

There are a couple of ramps in the initial setup that should never be visited (the only way to reach them is to send a 16th blue ball through) but aside from that and having to use the lone peg on the bottom row, it’s the most efficient possible way of counting with a 4-bit vertical register.


You’re right of course. It’s obvious on the simulator. With
real parts I had visually mistaken that the balls falling to the
right had to be kept away from the starting column.

As an improvement, if you replace the bottom two ramps
as I suggest, you can build the adder with only 3 additional
ramps. Based on your observation, the machine can be
improved to signal the “error” of feeding in too many
blue balls, by replacing one ramp with an interceptor.


“Use 15 balls or fewer” surely?


It should definitely be “fewer” rather than “less” since it’s answering a “how many?” question not a “how much?” question.

I haven’t been able to find an authoritative opinion for whether “15 or fewer balls” or “15 balls or fewer” would be more correct, but either would be more correct than any construction with “less”.

Checking further, the “N or fewer X” construction is in widespread use, so, in the absence of a rule against it, I’d accept it as valid.


“15 balls or fewer” sounds better than “15 or fewer balls” to my mind. However, at that point it becomes more of a style thing; i.e. subjective.


Yeah, for me, {15 or fewer} balls emphasises the significance of the precise value of the number while {15 balls} or fewer minimises the significance of the number so long as it’s not more than 15.