This week’s puzzle is an optimization problem: Can you determine the fastest way to toast and butter three slices of bread? (It’s harder than it sounds.)

## SUNDAY PUZZLE #35: Optimizing Toast

This week’s puzzle was suggested by reader Caroline S. Upon further investigation, I found that it traces back, as many puzzles do, to puzzle guru Martin Gardner. It’s a good one—we hope you enjoy it.

Even the simplest of household tasks can present complicated problems in operational research. Consider the preparation of three slices of hot buttered toast. The toaster is the old-fashioned type, with hinged doors on its two sides. It holds two pieces of bread at once but toasts each of them on one side only. To toast both sides it is necessary to open the doors and reverse the slices.

It takes three seconds to put a slice of bread into the toaster, three seconds to take it out, and three seconds to reverse a slice without removing it. Both hands are required for each of these operations, which means that it is not possible to put in, take out, or turn two slices simultaneously. Nor is it possible to butter a slice while another slice is being put into the toaster, turned, or taken out. The toasting time for one side of a piece of bread is thirty seconds. It takes twelve seconds to butter a slice.

Each slice is buttered on one side only. No side may be buttered until it has been toasted. A slice toasted and buttered on one side may be returned to the toaster for toasting on its other side. The toaster is warmed up at the start. In how short a time can three slices of bread be toasted on both sides and buttered?

We’ll be back next week with the solution—and a new puzzle! Got a great brainteaser, original or otherwise, that you’d like to see featured? E-mail me with your recommendations. (Be sure to include “Sunday Puzzle” in the subject line.)

## SOLUTION To Sunday Puzzle #34: Complete The Series

Last week, I presented you with a visual riddle the solution to which, I asserted, was not “6.”

As it turns out, this riddle has many correct solutions. As commenter Platypus Man put it: “There’s so little information [given], it could be just about anything if you properly rationalize it.”

By that perfectly sound logic, the answer could be, as Platypus Man proposes, five:

Put a horizontal line in the middle and you have three fractions. 1/2 to 3/4 is an addition of 1/4, so the last answer is 1. Given we already have the 5, the bottom number must be 5 as well.

It may not be the intended answer, but I don’t see how it’s any less valid.

The first row is inputs for a function f(n). In this case f(n) is the average of the first n twin primes minus one.

f(1) = 3 - 1 = 2

f(3) = (3+5+7)/3 - 1 = 4

f(5) = (3+5+7+11+13)/5 - 1 = 6.8

It could even be, as reader K. Praznik proposed by e-mail, 10:

(1*3) + (3-1) = 5 // (1*3) - (1-3) = 5

(2*4) + (4-2) = 10 // (2*4) - (2-4) = 10

So my answer to the sunday puzzle #34 is 10.

These solutions are all correct, but none of them is the answer I was looking for. The answer I was looking for was “R.”

“R” as in “reverse.”

That is, the pattern in which the numbers appear in the puzzle is the same as one commonly found on the gear levers of cars with five-speed manual transmissions:

#### Photo Credit: Zirguezi via Wikimedia Commons | Public Domain

That the answer was a letter and not a number was what I thought made this riddle “devious.” As it turns out, several of you had other, more colorful ways of describing it, which was fair; after all, it’s not like there’s a universally accepted gearstick layout (for five-speed transmissions or otherwise):

#### Photo Credit: dbkfrog via flickr | CC BY-NC-ND 2.0

You get the picture. The riddle was too open-ended. Whether you interpreted it as a mathematical puzzle, or an automotive design puzzle, it was poorly posed, and that’s on me. Puzzle-posing is an art in and of itself, and it’s easy to mess up. For a solution to be satisfying, the person posing the puzzle needs to provide enough information that the puzzle is unambiguously solvable, but not so much that it gives too much away.