After months of travel,
you’ve arrived at Duonia, home to the famous temple that’s
the destination of your pilgrimage. Entering from the northwest, you pass through the city gates
and the welcome center, where you’re given a map and a brochure. The map reveals that the town consists
of 16 blocks, formed by five streets that
run west to east, intersecting five more
that run north to south. You’re standing on the
northernmost street facing east, with the two blocks containing the gate
and the welcome center behind you. The temple’s only entrance lies
at the very southeast corner. It’s not a long walk,
but there’s a problem. As you learn from the brochure, Duonia imposes a unique tax
on all visitors, which must be paid when they arrive
at their destination within the city. The tax begins at zero, increases by two silver
for every block you walk east, and doubles for every block
you walk south. However, a recent reform
to make the tax fairer halves your total bill for
every block you walk north and subtracts two silver
for every block you walk west. Just passing through the gate
and the welcome center means you already owe four silver. As a pilgrim you carry no money
and have no way of earning any. What’s more, the rules
of your pilgrimage forbid you from walking over any stretch of
ground more than once during your journey— though you can cross your own path. Can you figure out a way to reach
the temple without owing any tax or walking the same block twice
in any direction? Pause here if you want
to figure it out for yourself. Answer in: 3 Answer in: 2 Answer in: 1 You look at the map
to consider your options. Walking towards the temple
always increases the tax, and walking away decreases it, so it seems like you can never reach it
without owing silver. But what happens when you walk
around a single block? If you start out owing four silver
and go clockwise starting east, your tax bill becomes six, then 12, then 10, then five. If you looped again, you’d owe seven, 14, 12, and six. It seems that each clockwise loop leaves
you owing one extra silver. What about a counterclockwise loop then? Starting owing four again
and going south first, your bill changes to eight, 10, five, and three. Looping again you’d owe six, eight, four, and two. Each counterclockwise loop actually earns
you one silver. That’s because any tax doubled, plus two, halved, and minus two, always ends up one smaller
than it started. The key here is that while the different
taxes for opposite directions may seem to balance each other out, the order in which they’re applied
makes a huge difference. You start off owing four silver, so four counterclockwise loops would
get you down to zero. Unfortunately, it’s not that simple,
since you can’t walk the same block twice. But there’s another way
to reduce your bill: walking one large counterclockwise loop
through the city. From your starting position,
walk three blocks south. You need to leave the southernmost
street clear for the final stretch, so continuing counterclockwise
means going east. Walk two blocks to the eastern wall
and you owe a whopping 36 silver. But now you can start reducing your bill. Three blocks north and one block west
cuts it to 2.5. You can’t go west from here
—that would leave you with no way out. So you go one block south,
and the remaining three blocks west, leaving you with a debt
of -1 silver. And since doubling a negative number
still gives you a negative number, walking the three blocks to the south wall
means the city owes you eight. Fortunately, that’s exactly enough to get you through the final blocks
to the temple. As you enter, you realize what
you’ve learned from your pilgrimage: sometimes an indirect route
is the best way to reach your destination.