Illustrator Martin Handford published the first in his beloved series of Where’s Waldo books over 25 years ago.* The books challenge readers to find the titular cartoon man, clad in his trusty red-striped shirt and red-striped hat, as he hides in a landscape of red-striped red herrings. When attempting to find Waldo you can scan the page completely from top to bottom, or you can focus your search around certain landmarks where Waldo seems likely to be hiding (in a castle’s moat, riding a blimp). Neither approach is particularly efficient. Which got me to wondering: What if there’s a better way?
I knew that Handford had placed Waldo in each of these illustrations, and in my experience, all people—even people who make a living hiding cartoon men in cartoon landscapes—have tendencies, be they conscious and unconscious. True randomness is very difficult to achieve, even if you want to, and according to Handford he does not necessarily aim for unpredictability. “As I work my way through a picture, I add Wally when I come to what I feel is a good place to hide him,” he once told Scholastic. Knowing this, is it possible, I wondered, to master Where’s Waldo by mapping Handford’s patterns?
I sought to answer these questions the way any mathematician who has no qualms about appearing ridiculous in public would: I sat in a Barnes & Noble for three hours flipping through all seven Where’s Waldo books with a tape measure.
The map born of my experiment is below.
It may not be immediately clear from looking at this map, but my hunch that there’s a better way to hunt was right. There isn’t one corner of the page where Waldo is always hiding; readers would have already noticed if his patterns were so obvious. What we do see, as highlighted in the map below, is that 53 percent of the time Waldo is hiding within one of two 1.5-inch tall bands, one starting three inches from the bottom of the page and another one starting seven inches from the bottom, stretching across the spread.
So, if you want to find Waldo on any given page, a good strategy would be to start by scrutinizing these two bands first, before moving on to other areas. Although 1.5 inches isn’t a particularly narrow range, it’s small enough to focus on without missing Waldo; and over half the time, he’ll be there. To test the efficacy of my approach, I pitted two Slate colleagues against one another in a Where’s Waldo showdown. L.V. Anderson searched the 11 spreads in the first Waldo book using the old-fashioned method of scanning the page; Dan Kois worked his way through the same pages, but armed with my findings (and a tape measure). Watch highlights from Dan’s decisive victory here:
Perhaps you’re wondering: Are these findings for real? It’s true that it is tempting to see patterns where they don’t exist, and it’s especially easy to do so when there are only 68 points to analyze; it’s a small sample size. But the probability of any two 1.5-inch bands containing at least 50 percent of all Waldo’s is remarkably slim, less than 0.3 percent. In other words, these findings aren’t a coincidence. Waldo is there for a reason.
(A quick note on methodology: The books I examined were the seven “primary books.” While Handford has churned out numerous puzzle books and other spin-offs over the years, the official Waldo canon, as I defined it, is comprised of these original seven volumes. Furthermore, the current editions of the first five Waldo books are “Special Edition” versions, first released in 1997. Waldo has been hiding where Handford put him in these “Special Edition” locations for the last 16 years. If you have a Waldo book on your shelf, chances are you have this version. So in conducting my analysis, I used this most recent hiding spot instead of the original ones. The majority of the series takes place on giant two-page spreads (roughly 20” x 12.5”) and only these pages were used in this analysis.)
So we’ve taken care of the question of where Waldo is. But that leaves a more intriguing question left unanswered: Why is Waldo there? Why, Waldo? Why are you so likely to hide in these two narrow bands? Why are you rarely at the edges of the page? Why are you rarely in the middle of the page? Why, Waldo?
I’d hoped to put the question to Waldo himself, or barring that, to Martin Handford, but the illustrator declined through his publicist to be interviewed for this article. That leaves me to postulate about Waldo’s proclivities.
Let’s start with the constraints Handford was working with in creating his books. Most of the two-page spreads have a “postcard” from Waldo in the upper left hand corner of the left page, which occupies a space of about 15 square inches. Waldo, clearly, cannot be hidden here. However, the corresponding space on the right hand page is home to Waldo only four times, slightly less than we would expect if Waldo were placed randomly. This suggests that the postcard is not depriving Handford of any favored territory.
Handford generally shies away from putting Waldo near the bottom or top of a page, which leads me to theorize that Waldo placement is largely a function of two factors: aversion from extremes and aversion from the middle. While we would expect Waldo to be hidden within an inch-and-a-half of the spread’s top or bottom borders almost 25 percent of the time if Handford were placing him randomly, in practice he is there only in 12 percent of all pages. More surprising is the fact that Waldo is also unlikely to be in the middle of the page, as you can see from the map above. It’s possible that Handford avoided the edges and centers of the pages out of concern that they may not print clearly. However, Waldo is placed in these locations occasionally, which weakens this hypothesis. I think it’s more likely that Handford was trying to avoid locations that might be construed as too obvious—i.e. the centers or the corners, where children and adults alike might begin their search. But while this might make Waldo harder to find for the reader whose eyes immediately dart to the center or edge of the page, once you know your quarry is unlikely to be in those places, it actually makes him much easier to find.
Correction, Nov. 19, 2013: This article originally misspelled the surname of Where’s Waldo author Martin Handford. The article also misstated the book’s debut as being 20 years ago. The first Where’s Waldo was published in 1987. (Return.)