May 30, 2021
Stuck on a problem? Look beneath the surface — you may have already solved it
A quick guide to problem solving by analogy
Here’s something to puzzle over.
A patient presents with a malignant, inoperable tumour. The doctor can use a particular type of ray to destroy the tumour. Unfortunately, the ray will also destroy healthy tissue in the process. The rays cause no damage to healthy tissue at lower intensity but, of course, they will not destroy the tumour at that intensity either. Can you describe a way to destroy the tumour without causing damage to healthy tissue?
We’ll return to our patient shortly. Before that, let’s play a game.
Sum to 15 is a two-player game in which players take turns to select numbers from 1–9. A number can only be chosen once. As soon as a player has three numbers whose sum is 15, they win. If no player can get to 15 in this way after all nine numbers have been chosen, the game ends in a tie.
Can you come up with a winning strategy?
Here are three run-throughs of the game. Before scrolling any further, follow them turn-by-turn and see where your intuitions take you.
Does this game have an air of familiarity to it? There is a strong element of ‘blocking’ — choosing a number that will prevent your opponent from getting to 15 (in the second game, for instance, every move from the fifth onwards is a block). Player 1 seems to be at an advantage, but games can end in a tie. And the all-important rule involves a configuration of three items.
We’ve seen this before
As we continue to play, it dawns on us that sum to 15 bears the hallmarks of that well-trodden game, tic tac toe (or noughts and crosses, as it’s known to some). This isn’t immediately obvious — the rules of sum to 15 seem to conspire to turn a game involving numbers and addition into a more abstract game that has no numbers in sight.
Mathematicians learn to abstract away from the surface-level details of a problem (like numbers and addition) and extract its essential structure. They are constantly on the lookout for connections between seemingly disparate ideas.
To make the connection explicit in this case, we need to represent sum to 15 as a 3x3 grid. The winning ingredient in this game, remember, is that three numbers combine to 15. So we’re looking for a way to present the numbers 1–9 on a 3x3 grid in such a way that every row, column and diagonal sums to 15 (and accounts for all possible sums to 15). This is nothing other than the famed magic square!
The starring role of magic squares in sum to 15 may not have come to mind right away. But as we play through a few times, and tease out the similarities to tic tac toe, the squares emerge as a bridge between the two games.
We can now formulate our strategies for sum to 15 in terms of everything we know about tic tac toe. The middle space has the most routes to victory (equivalent: 5 is involved in the most combinations to 15). If your opponent starts in the middle, choose a corner (equivalent: if your opponent starts with 5, respond with an even number). Most importantly, if both players exercise perfect strategy every game ends in a tie.
Just borrow the solution
This hopping between representations is a common strategy of mathematicians — ‘solving by analogy’ allows us to tackle one problem by framing it in terms of structurally similar, even identical, problems we’ve already solved before. Our success rates on problems increase manifold when we’re able to look beneath their surface.
Back to the patient — did you figure out a safe method of destroying the tumour? If not, you are in the majority — only 10% of people generate a valid solution. 1
Now allow me to tell you about a General who is seeking to capture a fortress that happens to be located in the middle of a country ruled by a dictator. There are several roads leading to the fortress through the surrounding countryside. The General knows that the dictator has peppered all of those routes with mines, and that if any large force descends on any one of those routes, they will detonate the mines. Rather than send his forces along a single route, the General divides them into smaller groups that each take separate routes. They can each make their way to the fortress safely and arrive at the same time.
You may know where this is going. Can you solve the radiation problem now?
The success rate triples to 30% for participants who are shown the fortress story. When they are explicitly told that there is a link between the two, the success rate rockets to 92%. The fortress story is, of course, just the radiation problem in disguise; the General’s dilemma, and his solution, maps over perfectly. The doctor just needs to direct multiple low-intensity rays from different angles.
As a standalone problem, the radiation problem requires a creative insight that eludes most of us. Taken at face value, we may mistakenly assume it requires deep medical knowledge. But when we probe beneath the surface of the problem, we can relate it to others with the same deep structure, whose solution has already yielded to us. At that point it’s no longer a matter of inventing a solution as much as borrowing a preexisting one.
Problem solving is largely a conscious exercise in analogy (conscious because if we’re not looking for linkages they may escape our attention). The epiphanies of famed geniuses — whether the fictional ones that anchored each episode of House or the real ones of math olympians — are often rooted in things they have seen (and solved) before.
The Hungarian mathematician Paul Erdös once quipped that ‘a problem worthy of attack, proves its worth by fighting back’. I might dispense with the poetry to add that it also contains interesting structures that recur, often in subtle ways, in later problems. The more of these problems we engage with, and the more attention we pay to their deep structure, the more we can lean on them as we seek that critical breakthrough.
A puzzle a day keeps your future struggles at bay, perhaps?
