Stephen Sondheim adores cryptic clues—or British-style clues, as they’re called in his neck of the woods. The American composer puts it this way: ‘The nice thing about doing a crossword puzzle is you know there is a solution.’
Eventually, presuming you persevere. Because some solutions don’t come fast; you need to stalk them, shake them into sight, presuming an answer comes at all.
Or you surrender. I mean, why not? Waving the white flag requires a lot less strain than torturing the neurons for an hour. If the puzzle is part of a book, flick to the back to check the answer. If the puzzle appears in the newspaper, wait until tomorrow to fill in those blank squares.
Yet stubborn solvers don’t, and one big reason why is called the ‘aha’ rush. You know the answer exists, so you persist. You scratch a little deeper until you relieve the itch.
Take a clue from Joon Pahk, a freelance compiler for the New York Times, the city that’s home to Sondheim. Pahk’s puzzle ran a few years back, including this innocuous clue:
Number of holidays?
The answer has five letters. At the time, trying to fill the grid, I suspected THREE was the answer. Or SEVEN. How many holidays in a calendar year? EIGHT?
Or maybe the clue had a sardonic edge, prompting the likes of ZILCH or I WISH. (Puzzles in America don’t give an answer’s letter count, aside from the number of squares allocated within the grid. Nor do US clues indicate if the answer is a phrase, entailing several words. Hence I WISH or IN TOW might occupy five blanks.) Then again, perhaps QUOTA was the so-called holiday number, one’s allotted rest-days in the working year. Even LEAVE seemed plausible, the number of holidays you accrue. By this stage you’re right to suspect the clue was innocuous only on the surface.
As much cryptic as quick, Pahk’s wording aims to sidetrack your brain, sending it down the wrong path. Either you surrender and check the answer, seeing where you were fooled, or you hold your nerve, knowing the blissful aha rush will reward your effort.
Cryptic crossword solving comes down to faith on two levels. First your faith in the setter, believing their solution will be worth the trouble of seeking. And second, the faith in your own neurons, trusting that the answer can be summoned from your neural HQ. To fuel this twofold faith is the prospect of a ‘eureka’ moment. Even if you’ve never solved a crossword, a kenken or a logic puzzle, I guarantee you’ve already experienced the glee a breakthrough brings. You face a problem; you reach an impasse; the answer seems unreachable, until you rethink the situation and bang—the high arrives. Not only does a problem get fixed or a puzzle completed, but your brain feels fulfilled.
But why? What triggers the brain’s sudden insight, and why does the brain seem so charged when that eureka finally arrives?
It’s time then to examine this aha moment—that occupational pleasure among solvers—as well as the genesis of insights in general. Facing a dilemma or a difficult clue, how does our brain turn a brick wall into a light bulb?
And if you’re still not sure about Joon Pahk’s clue, don’t fret. The answer will arrive before this chapter’s end, whether by my hand, or by your grey matter. But first, to understand aha, we need to play with matches.
Houston, we have a problem
One day in Texas, Bhavin Sheth was playing with matches. Less in a pyro way than a neuro way. The plan was to use the props as a means of measuring brainwaves.
Dr Sheth, an associate professor in neuroscience at the University of Houston, was using a series of puzzles to monitor how solvers solve, to chart how neurons work under pressure, and trace the buzz of that glorious aha.
His guinea pigs were students selected from campus. The puzzles were a series of cognitive tests, from lateral to lexical, from easy to unorthodox, just like the matchstick puzzle below. On the table, the equation didn’t make sense:
Translated from the Roman numerals, the sequence reads:
11 + 1 = 10
Which it doesn’t. Even a toddler could tell you that. Opening the way to Dr Sheth’s question: how do you correct the sum, moving as few sticks as possible, and still keep 10 as the solution?
The crunch is moving as few sticks as possible. The obvious remedy would be to simply move the second match, transforming the equation in one tweak to:
It’s neat, but Sheth encouraged the students to look harder. Smarter. There was a simpler way, he promised. To measure how the solvers coped, Sheth harnessed each student to an electroencephalogram, or EEG. Imagine a hairnet made of cables, where each cable links to a pair of electrodes placed on the scalp. As a pair, the electrodes will vary by what voltage they receive via the brain, this fluctuation charted in peaks and troughs on the screen. Professor Sheth studied these waves as his students wangled matches.
And wangled. And wangled. They tried a dozen approaches—altering symbols, turning Xs into Vs, editing the maths symbols—but none could imagine how anything could be more minimal than the single move shown above.
This narrowness of thinking is a symptom of functional fixedness, as the mindset is known. Too often we observe things in a prescribed light, as if one scenario demands one outlook, and one outlook only. Such thinking forfeits the potential of multiple perspectives, stuck in one groove at the expense of smarter ways.
For example, if I write the word FLOWER, what do you think of? Possibly blooms spring to mind—or spring itself. You might picture daisies and tulips, wreaths and bouquets, kindergarten drawings or an ikebana bowl. Yet how many people entertain the idea of something that flows, literally a FLOW-ER?
That is fixedness at work. We cling to unique interpretations despite others existing. We prefigure an answer and make every effort to reach it, regardless of other potential answers. In the same way, we’ll gaze at 9, the figure, seldom conceiving how the same squiggle can represent 6 with a twist of the wrist. Thanks to fixedness, we think too heavily along one tangent, deepening a rut out of routine.
Concentration like this can be both blessing and curse. Solvers need to focus to fix a problem, yet not fixate to the point of lethargy. Sri Lankans know the syndrome as kupa-manduka, or frog in the well, where a stricture of thinking equates to a narrowness of perspective, just as the frog in the well can only observe a tiny circle of sky rather than the sky’s true width.
The idiom holds true in problem-solving. Stare too hard at any conundrum and you’re likely to be blinded by a single viewpoint. Even the word concentration implies a clustering, the word implying a pooling of perspective, a centralised bunching of resources, as if every conscious thought is jammed through one funnel.
Back in Houston, Bhavin Sheth observed the students’ brainwaves were dominated by gamma waves, the common ‘tell’ of concentration. The gamma pattern calls the brain to attention, sweeping the lobes front to back some forty times per second, the rhythm depicted on the EEG’s screen.
For all the focus, however, the students were getting nowhere. Did they read the instructions correctly? Quote, unquote: can you correct the sum, moving as few sticks as possible, and still keep 10 as the solution?
I’ve already given you several hints about the answer, but if you need a little more encouragement, I vote we switch from matches to music.
Jazz detour
Dr Charles Limb, jazz musician and surgeon, is based at the University of California. He likens our brain to the cosmos, an inner space no less bewildering than the galaxies beyond us. One key tool to help us grasp this neural universe, the field’s own telescope in a sense, is the fMRI tube.
Several steps advanced from an EEG, functional Magnetic Resonance Imaging relies on blood flow. An active bit of brain needs more oxygen, delivered by blood, than a less active brain portion. In essence, the fMRI detects these distinctions, charting where oxygen is in peak demand, and which parts of the brain are relatively idle. While the fMRI still has drawbacks—its din, the terror its confines hold for claustrophobics—Limb deems the tube the best tool in the shop, a prized means of mapping our brains’ points of light, just as ancients had charted the constellations.
Rather than puzzle-solving, Limb’s focus was on improvisation, inviting jazz musicians to carry a tune into score-free territory. ‘Artistic creativity—it’s magical, but it’s not magic,’ as Limb put it. ‘Meaning that it’s a product of the brain.’ As both scientist and muso himself, Limb was intrigued to see the mental genesis of creative solutions.
He was using the method known as BOLD imaging, or blood-oxygen-level dependent imaging. Whenever a neuron needs energy, it draws oxygen and sugar from the blood. This transaction is captured by the fMRI, allowing scientists to see which lobes are doing the greater toil, almost like a traffic update for a metropolis.
Which precinct then would host the heaviest ‘jam’ when it came to musical jamming? That was the question to test as Mike Pope, a jazz guru with a zeal for improvisation, squeezed inside an fMRI tube in Dr Limb’s clinic. Tightening the squeeze was a toy-sized keyboard laid across Pope’s thighs, a baby piano with special powers: instead of producing notes, the keyboard absorbed finger pressure. Pope was asked to respond to some jazz, a series of exploratory riffs from Limb himself on an orthodox keyboard outside the scanner.
MATCH SETS
Will your neurons be nimble enough to solve these six spatial puzzles? The solutions are below.
Puzzle 1
Remove three matchsticks to make the equation true.
Puzzle 2
Here are six matches. Add five more and make nine.
Puzzle 3
Move three matchsticks to make three equilateral triangles.
Puzzle 4
Can you move three matches so that there are four triangles?
Puzzle 5
Move four matchsticks to make three squares.
Puzzle 6
Here are two wine glasses, built from ten matchsticks. Can you move six of these to build the house that holds the glasses?
Picture the scene, a jazz club in a clinic, musicians in lab coats. Limb played four bars as a stimulus for Pope’s response, the subject fluidly pressing his keys inside the tube, replying with silent music. To sustain the interplay, Pope’s music was instantaneously translated into audible phrasings via software so that Dr Limb could respond, and so on. The two men traded music with abandon, winding between minor and major keys, generating ahas aplenty as the melody wandered. But instead of sharing a stage, half of the duo lay prone in a magnetic field, his every neural flare-path logged for posterity.
The initial data sketched an astonishing picture. Two things in particular caught Limb’s eye. The first was the intense activity in ‘Broca’s area’, a pocket in the frontal lobe named after Paul Broca, a French physician who plumbed the mysteries of speech production 150 years ago. This indicated music is a grammar as much as any other tongue, a semantic system of octaves and semiquavers.
The second revelation related to darkness. Our metaphors for aha moments are insistently suffused with light—a bulb ignites above our head, scales fall from our eyes, allowing us to see the light, the truth. Yet Mike Pope’s brain-map told a different truth. Throughout the riffing, the subject’s medial prefrontal cortex was ablaze. This brain segment, level with the hairline, is linked to decision-making: a driver reflexively choosing a freeway lane without conscious thought, the solver’s impulse to move a match. But as jazz filled the clinic, Pope’s lateral prefrontal cortex was largely idle, a car sitting in neutral. This region is central to introspection. If the BOLD screens were any guide, Pope was hardly aware of being Pope as he played. His self-reflection was all but asleep, the blankness all the more dramatic when compared to the fireworks in the adjacent region, the medial prefrontal cortex vested with self-expression.
MATCH SETS
Did you manage to match wits?
Puzzle 1
Thanks to Matchstick Puzzles at <http://matchstickpuzzles.blogspot.com.au>
Puzzle 2
Maxey Brooke, Trick, Games and Puzzles with Matches at <www.arvindguptatoys.com/arvindgupta/matchplay.pdf>
Puzzle 3
Puzzle 4
Designed by the Grabarchuk family, <http://grabarchukpuzzles.com> via <http://matchstickpuzzles.blogspot.com.au/>.
Puzzle 5
Puzzle supplied by Mike Daigneault via <www.learning-tree.org.uk/stickpuzzles/stick_puzzles_old.htm>.
Puzzle 6
Courtesy of <www.puzzles.com/PuzzlePlayground/TheWineGlasses/TheWineGlasses.htm>.
Hard conclusions are yet to be drawn from Limb’s work, but the inference is clear. In order to create new solutions, some neurons need muzzling as much as others need unleashing—introversion must give way to the daredevil who doesn’t blush at making mistakes. Find the off switch in one brain zone and you empower the other.
But how do you do that? If you wish to arrange notes, or grapple with crossword clues, how can you get your neurons to cooperate? Let’s go back to that problem in Houston, with what we know now.
Perfect matches
Any luck with those matches? Or the so-called number of holidays? Keep the jazz musicians in mind, the way Mike Pope’s brain suppressed its executive control to access a wilder sense of play. If the muso made a blunder, pressed the wrong key, the melody swept along regardless. There was no harm in thinking loose.
Back in Houston, Bhavin Sheth noted the more that gamma waves ruled their brains, the further the students stalled—their matches intact, the elegant solution unseen. XI + I = X seemed etched in stone. We know some tried to move a match, making IX + I = X, which was admirable, but even that tweak could be surpassed. Yet how?
My first hint came a short while back, talking about 9 becoming 6 if you adopted a novel viewpoint. The second hint was the jazz experiment, seeing two musicians quit the score and treat music as an open space to move within. The technical term is restructuring, where your brain relents its fixity. In a sense, you sublimate the problem, placing the focal point offstage, out of sight, yet keeping the puzzle’s challenge active in your working memory. You look elsewhere, in a way. Think otherwise. Because to experience that elusive aha, you need to be digressive as well as obsessive.
You must concentrate—then deviate. I’m sure you’ve noticed how many great ideas emerge out of daydreams, a dog walk, a prolonged shower, when daily cares are suppressed, even as your subconscious quietly toils on. The archetype of this is Archimedes, the genius lying back in his bath. The moment the water rose as he sank, the Greek happened upon the law of displacement, a problem he’d been gnawing at for ages. His resting brain solved what his fixated brain could not.
The students of Houston were nearing the same moment. Staring at matches can only get you so far. Besides, the matches were not the ones in need of changing. The equation could stay intact on the table. Only the observers were obliged to refresh their perspective.
Eureka. A few seconds before each solver solved the matchstick puzzle, a calmer theta wave pattern washed over the screen and the gamma waves dissipated. Compared to the busy gamma undulations, theta is more a level sea—the calm tidal tempo often bearing the insight that’s been elusive for so long.
Along with the shift in wave patterns, Sheth observed a spot fire deep in the brain on the EEG monitor. This neural activity centred on the limbic cortex, the centre of emotional regulation lying astride the thalamus.
But that wasn’t all. Thanks to BOLD imaging, Sheth could see a disco-like flashing in the anterior cingulate area of the limbic cortex, a band that fringes the corpus callosum. The anterior cingulate is deemed an important zone for impulse control, as well as the precinct that anticipates reward. In effect, the brain was changing its framework, relinquishing focus for freestyle thinking.
Houston, we have a solution. The answer to the matchstick problem relied on inverting the equation, or changing sides at the bench.
This was the rethink: finding a visuospatial solution to a mathematical problem. Realigned, the new sum asserted X = I + IX, which was true, the total unchanged, and not one match in need of moving.
Which leaves us with one more puzzle to solve, namely Joon Pahk’s clue: number of holidays? Any theories? These stories of improvised music and inverted matches should urge you to read the words in a different light. Number can mean several things apart from 1, 2, 3.
Just as flowers can flow, numbers can numb. That was my eventual response when solving Pahk’s puzzle, entering the all-numbing BOOZE into the squares. Alcohol is a popular source of anaesthesia after all. But why did Pahk specify the holidays? That element didn’t quite gel, goading my brain to stray in a new direction even as I ventured deeper into the crossword. When a C arrived as the answer’s initial, I toyed with COUNT. Or maybe CROWD—streets are often swamped by extra numbers during the holidays. But then I saw the light—or the darkness set in.
I imagine that gamma waves stilled in tandem with the dimming of self-awareness, a muting of that insistence upon one interpretation … I read the clue as if for the first time. My mind switched over. The narrow-sighted frog fled the well. Suddenly ‘Jingle Bells’ rang a different bell, as I realised holidays referred to a particular holiday, and number was neither integer nor intoxicant, but something closer to a jazz tune. Number of holidays—the Christmas holidays, to be exact—is a CAROL.
These puzzles—a spatial trick and a sly piece of wording—channelled so many brains in so many false directions. In the end the answers lay in the art of fresh thinking, turning the tables, widening the frame. Yet that’s just part of the cognition story. To learn a little more, to see the aha itself from a new perspective, let’s look at tempo, and see how puzzles help the brain shift gears.
