While at 15Five, we learned the concept of breakthrough thinking. It was a buzzword of sorts, and it took a while to understand what it meant. It's one of my favorite things I took away from working there. Thanks, Jon.

Breakthrough thinking is a beautiful concept. The basic message is that we work within bounds when we try to solve a problem. Working within bounds is natural, as every solution must fit within some limits and constraints. However, we impose some of the bounds ourselves, preventing us from coming up with great solutions. We do it without thinking. Very often, we're not even aware of it. Even when someone points this out, it causes discomfort and resistance. It illustrates how we are often the ones who are standing in our way.

While breakthrough thinking seemed simple initially, I did not grasp it immediately. The word is easy enough to understand. It's "thinking outside of the box" using different words. It took a while longer to understand what it really means and recognize opportunities to use it.

Recently, I had a lovely experience that helped to explain the concept of breakthrough thinking. Read on unless you hate riddles or don't want spoilers.

⚠️ Warning: riddles with spoilers ahead. ⚠️

On Saturdays, we have a big family breakfast in our house. Today, my kids decided to make avocado toast with poached eggs. They were buzzing because they came up with this idea and prepared breakfast all on their own. The food was yummy, and everyone was in a great mood.

My 10-year-old daughter asked a riddle. "What grows with its roots pointing upward." We racked our brains. Soon enough, someone guessed correctly: "Teeth in the upper jaw!" We laughed at how simple the answer is once you hear it.

We continued. I followed up with, "I went to the doctor today. The doctor said, 'Son, you've got a cold, but this doctor was not my father." My kids are used to dad jokes and suspected a trick. "Your name is Son," said my 8-year-old son. How well do they know me! Good try, but not the answer. "The doctor is a priest," said my daughter. Good one, but not it. The doctor was my mom.

Their faces brightened up. We were on a roll. My wife went next: "How can you cut a cake into eight pieces using only three cuts?" The kids were puzzled at first but eventually figured it out. The first cut divides the cake into two half-circle halves. The second cut divides the cake into quarters. The third cut goes sideways and divides the cake into a top and a bottom part. You end up with eight slices. Brilliant!

We were delighted.

Alright, I said. Cutting the cake was easy, but you could not do that with a pizza. Let's try cutting a pizza instead. Here's the challenge: cut a pizza into seven pieces with only three cuts.

At first, the kids did not get it. The looks on their faces, as they were thinking, were beautiful. Then, someone figured it out: "The pieces do not have to be the same size! You do three cuts, but not through the middle".

We're on a roll. "That was easy, I said. Try this. Cut a round pizza into eight equal pieces using only three cuts." The kids were stumped. They tried to argue that you could do the same thing as with the cake earlier, but we realized that in practice, that won't work. I assured them that there was a better and simpler solution. They just had to think more open-mindedly.

They gave up. The answer is that you first cut the pizza in half. With the second cut, you cut it into four quarters. Then you stack the quarters on top of each other, and with the final cut, you end up with eight equal slices. Sure, it will make a mess, but no one ever said it had to be neat.

They were buzzing at this point. My daughter then said she could slice a pizza into eight pieces with only two cuts. Once again, no one could figure it out. She explained that the first cut would be a wave, creating two pieces that look like the black and white parts of a yin-yang (without the little circles). The second cut would be a straight line, creating eight very uneven pizza pieces.

We soon realized you can cut a pizza into an arbitrary number of pieces in one continuous cut! The possibilities are endless as long as the cut can curve and form loops. It was an excellent conclusion to a fantastic train of thought.

That, in a nutshell, is breakthrough thinking. In each of these riddles, the only thing preventing you from coming up with the answer is an unstated assumption.

• With the roots, one of the assumptions is that they are plant roots. Another one is that they will be visible and exposed since they are pointing upward. Teeth have roots, and in the upper jaw, they point upwards without being visible!
• With the doctor riddle, we are conditioned culturally to assume that doctors are men.
• With the three-cut cake problem, the assumption is that the cuts must be made on the same plane. Cutting through the cake sideways solves it.
• The seven-piece pizza with three cuts problem tricks people because they think pizza and think that the resulting pieces must be equal.
• The 8-piece pizza with three cuts assumes you don't move the slices between cuts.
• The 8-piece pizza with two cuts is hard because people assume the cuts must be straight. The final breakthrough came when we realized that a single cut can curve and loop back over itself.

In each of these cases, the assumption is in your head. It is implicit. It was never stated. It comes from experience or because things are usually done that way.

These are the things that make a good riddle. The solution is simple once you see it. It involves something surprising, which is pleasing.

Breakthrough thinking is like that. It's a way to expand your thinking and recognize unstated (and sometimes stated) assumptions standing in your way.

The first step is to ask yourself what assumptions, limitations, and boundaries you have in your head. Then, you ask yourself which are stated and which are implicit. Next, start exploring what solutions are possible once you remove one or more of those implicit boundaries. Finally, ask yourself if removing or relaxing the boundary is acceptable.

1. Recognize limits, boundaries, and assumptions that are shaping your solutions
2. Questions which of these are implicit, self-imposed, or explicit, but negotiable.
3. Explore what solutions are possible if you remove or relax one or more of these limits, boundaries, or limitations.
4. Ask if removing or relaxing them is acceptable.

Incredible things happen when you apply this to real-world problem-solving. First, you realize that many limitations are entirely arbitrary. They are only there because that is always how it was done. All it takes is to realize that, question it, and all of a sudden, new possibilities arise. You can feel that moment. Eureka! It's a breakthrough! That's why this is called breakthrough thinking.

Not all boundaries are self-imposed or implicit. Some are explicitly stated. Breakthrough thinking also leads to questioning external assumptions and boundaries. That is perfectly OK. It leads to conversations and negotiations between stakeholders.

If you're asked to do something, ask the other person, "Would it be acceptable if I did X while doing it?" or "You said that it should not do Y, but what if it did it just a little?". When you ask things like that, the other person must reevaluate their assumptions and also use breakthrough thinking. Such conversations can lead to novel solutions. Sometimes stated boundaries need clarification, are not what they seem, or can be negotiated.

As a parting gift, here's an eye-opening article about things you are allowed to do. It's fun because some of those things might not have crossed your mind but are perfectly acceptable and can help you uncover new insights and accomplish new things.