When my 12-year-old nephew invites me over, it can only mean one thing — he needs help with his maths homework.
The problem seemed simple enough:
Take four copies of the number 1234. Re-arrange the digits in each number so that the four numbers sum to 9000.
For instance, taking the numbers as is gives 4936:
By reversing the digits in the first number, we can get closer to the target:
The challenge is to hit 9000 dead-on.
I was surprised. My nephew is in Year 8 and punches above his weight in maths. I have seen him rattle off multi-digit addition problems with maddening accuracy. Why had the solution eluded him?
I scribbled away, beginning with trial and error before turning my focus to the ones column. That already narrows the choices, because the four digits in this column must add up to 10. Regardless of how I rearrange the four numbers, the final digits must comprise 4,4,1,1 or 3,3,3,1 or 2,2,2,4 or 2,2,3,3 or 4,3,2,1. I continued down this path, hoping to systematically arrive at the solution. It was exhausting work and I could see why my nephew was struggling — there is no obvious combination that works, and there are a lot of possibilities to burn through.
After 15 minutes, a new possibility dawned on me. There might not be a solution. I redirected my mental energy in search of a mathematical argument — a proof — that 9000 could not be reached.
I went back to my early attempts. Suppose we have a combination that does sum to 9000. Then, using basic properties of column addition:
- The four digits in the Ones column must add up to ten.
- For similar reasons, the four digits in the Tens column must add to 9. Same for the Hundreds column.
- The Thousands column must add up to 8.
Taken together, this means that all of the digits in the solution would have to add up to 10+9+9+8=36.
But…they can’t. Look at the numbers again: four copies of 1234. In each copy, the digits add to 10, making a grand total of 40. However we rearrange the digits, they will still sum to 40, meaning we can’t reach our target of 9000.
Pretty cool, right? My nephew certainly thought so. As I guided him through the proof, I could detect his satisfaction — there’s nothing quite like an ‘aha’ moment. I had him explain the reasoning to his mother and left knowing he’d grasped the main point — there is no solution.
My work here was done — or so I thought.
What happened next was revealing. My sister told me that my nephew devoted the rest of the evening to searching for a solution. That’s a strange commitment to make when you’ve just been convinced that no such solution exists.
My nephew was in a state of cognitive dissonance. He understood that this problem has no solution. And yet, this was his maths homework. From all his years of schooling, every maths question always had an answer. Why should this time be any different?
My nephew’s deference towards school maths, with closed questions and prescribed answers, betrayed his mathematical reasoning.
He ultimately could not bring himself to accept that a solution does not exist — to him, that was not how mathematics operated. And while his perseverance that evening was laudable, it was also misplaced.
AAfter some digging, I learned that my nephew’s teacher was indeed aware that no solution exists. He selected the problem as an opportunity for his students to apply their fluency and understanding of column addition within a novel context. What an inspired choice it was.
The execution, however, was poor. There was no follow-up to this ‘extension problem’ — no feedback or discussion. It was only through my nephew’s dogged persistence (mine too, admittedly) that this issue was even raised. By then, the class had moved on to the next topic.
I wonder how many students were left frustrated at having not found a solution they assume must have existed.
School mathematics is presented as a collection of immutable truths. Problems are binary: there is a right answer and you’re damned if you can’t find it. Curriculum standards do not reflect the rich, exploratory forms of problem solving that underpin mathematical thinking.
Students deserve to be exposed to these problems because they animate the uncertain and often surprising world of maths. They hold the key to unlocking creative thinkers who embrace the unknown and break free of fixed ideas. This is the maths students will need beyond school.
Open-ended problems are the lifeblood of mathematics — we cannot afford to relegate them to a sideshow.
