# The Perils of Puzzling: Alternate solutions!

[Thinking hard. Image courtesy of popsci.com]

The science, fantasy, and science fiction website io9 has a marvelous weekly feature run by Robbie Gonzalez, wherein they tackle brain teasers and riddles both new and old. I’ve explored several of them here on PuzzleNation Blog, most notably the 100 Men in Hats puzzle, which expanded on the Men in Hats puzzle concept from one of our earliest posts.

But one of their latest riddles provided a valuable example of how crucial test-solving and crowd-sourcing can be to a puzzle’s success.

The idea was simple enough: look at the numbers below, and determine what number should take the place of the question mark. The only guideline? The answer was NOT six.

I posted this riddle on our Facebook page on Monday and shared it with fellow editors at the PuzzleNation office, and got all sorts of answers in return.

One solver came up with 5 as the answer, positing that the vertical numbers formed fractions. So, with 1/2 and then 3/4 as the next number, the pattern would be adding 1/4. Adding 1/4 to 3/4 equals 1, and 5/5 equals 1.

There were other solutions that also yielded 5 as an answer, like doing what my friend called a zigzag equation, adding 1 from the top to 4 from the bottom to get 5 on the top as the answer, and then reversing it by adding 2 from the bottom to 3 from the top, getting 5 on the bottom as the missing answer.

A second solver came up with 3 as the answer, adding the top row to equal 9, and then trying to do the same with the bottom row.

Another solver saw them as two separate patterns, where going from 1 to 3 involved adding 2 and going from 2 to 4 involved multiplying by two. Therefore, by this method, the answer is 8. (Yet another solver did the same, except they squared the numbers along the bottom row, leading to 16 as the answer.)

As you can see, there were all sorts of mathematical solutions. When you’re told to ignore the most obvious solution, your mind can create some truly innovative ways of reimagining the information available.

[A head full of numbers. Image courtesy of equip.org]

Several solvers thought outside the box and came up with R, relating the numbers by their positions on a gearshift knob instead of mathematically.

As it turns out, this was the solution the puzzle’s creator initially intended, only realizing later that the puzzle had many possible solutions.

In his own words: 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.

[A proposed layout that points more directly toward the creator’s intended solution.]

Now, as a puzzler myself, I can absolutely empathize with Mr. Gonzalez here. There are plenty of times I’ve created a puzzle or a brain teaser and assumed that everyone would follow the same path I envisioned, considering the solution if not obvious, then at least reproduceable.

But solvers can always surprise you by finding alternate routes to the answer or utilizing a different way of thinking that ends with a second, but still valid solution.

So after a few stumbles and missteps of my own in the past that were similar to the one in today’s puzzle, I now make sure to have another set of eyes on my brain teasers, either during the creation process or as a test-solver afterward.

A second set of eyes can be absolutely invaluable in helping you spot possible alternate solutions.

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# Let’s Make a Deal!

It’s a scenario every game show fan knows well. You’ve got three doors to choose from, and one of those doors will open to reveal a fabulous prize.

After you’ve made your choice (let’s say Door #2), our affable host Monty Hall plays Devil’s Advocate by opening one of the doors you didn’t choose (let’s say Door #1), revealing a goat or other lackluster result.

And then, Monty offers you a chance to change your mind. Will you stick with the door you initially chose, or will you switch to the other unopened door (Door #3)?

The average player sees two choices, Door #2 and Door #3, which on the surface sounds like a 50/50 shot, a coin flip. So would it surprise you to learn that people who switched from one door to the other doubled their chances to win the fabulous prize?

This is known as the Monty Hall Problem, an example of how statistics aren’t always what they seem, and it has puzzled people for decades.

It’s counterintuitive, isn’t it? I mean, you have two choices, so the odds should be 50/50. But you’re forgetting that third door that Monty eliminated. That third door makes all the difference, statistically speaking.

Let’s break it down. Your initial choice is between 3 doors, meaning you have a 1 in 3 chance of picking the correct door, and a 2 in 3 chance of picking the wrong one.

When Monty opens that other door, the odds haven’t changed. Only the number of options available has changed. Your door is still a 1 in 3 chance of being correct and a 2 in 3 chance of being wrong. But the remaining door now has a 2 in 3 chance of being correct!

So what appeared to be a coin flip between sticking with your choice and switching is now heavily weighted toward switching!

There have been several real-world tests of the Monty Hall Problem, and all of them have consistently shown that the people who switch were twice as likely to open the winning door!

The real puzzle here is how we fool ourselves. We take the numbers at face value — 3 doors become 2 doors, so a 1 in 3 chance becomes a 1 in 2 chance — and actually hurt our chances with those seemingly simple assumptions.

Being able to reconsider your assumptions is a major tool in the puzzler’s solving kit. Plenty of tricky crossword clues depend on you associating the clue with one thing, when the answer is something quite different.

After all, if you saw the clue “Unlocked” for a four-letter entry, you’d probably try OPEN before you tried BALD. Clever constructors are counting on that.

So be sure to remember Monty Hall and his three-door conundrum the next time you’re stumped on a puzzle. Maybe the answer is as simple as trying another door.

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