You, a detective

The mindset you enter a problem with opens or shuts doors of perception, and what you see or can't see is dependent on the climate of thinking you're in before you even begin the problem.

With that understanding, one of my favorite hats to put on to solve math problems is my detective hat, but not just any detective.

Instead of looking for the "right" answer, I'm looking for the path to the answer that feels the best for me.

In other words, I take all the pressure of performing off of myself, because my attention is on the process, not the outcome.

Let's take a problem like this:

374 x 221.

I could just multiply it out vertically and crosswise.

But if I'm especially mindful I may notice the number 374 is a multiple of 11.

It must be because any 3-digit number whose middle term is the sum of the outer terms is by definition divisible by 11.

If I were racing the clock or putting pressure on myself to get the answer right, the chances I would notice such a subtle clue plummet.

It's only when I'm relaxed any really noticing what's happening, any new thing about the problem, that my mind turns on, that I'm engaged and here to explore.

So how does that make the problem feel better? How does that make things easier?

Let's find out, together.

To divide 374 by 11, I simply remove the middle term, for 34. That makes sense, because if the middle term was generated by adding the outside terms to multiply by 11, removing the middle term must be the same as dividing.

So now I have 11 x 34 x 221.

But we can do better.

221 is not even, nor is it divisible by 3, because the digit sum of 221 (2+2+1) is 5, and in order for a number to be divisible by 3, its digit sum must be divisible by 3.

With this reasoning so far, I can't be sure that 221 is divisible by 3, but I know that it would be much easier to tackle the other two figures, 11 and 34.

So I could multiply 34 and 221 with vertically and crosswise, which is already a huge improvement because now I am multiplying with digits all less than 5, which is very easy.

So:

221

x 034

Vertically and crosswise I would have:

2 *0 = 0

2*3 = 6 + 2*0 = 0.

So far then we have 06 left to right.

Then 2*4 = 8 + 1*0 = 0 + 2 * 3 = 6, and so adding the 8 and 6 we have 14. the 1 from the 14 is carried back one column so now left to right we have 074.

Then we have 2*4=8 and 1*3=3 for 11. Now left to right, the 1 from that 11 gets carried and we have 0751.

Finally on the right we have 4.

So 07514.

Now I still have that 11 to multiply by it's easy because have an algorithm for it.

82654.

So that was very easy to do mentally, and I'm pleased with it.

That's the key part, is it feeling good to me? Much more important than speed, because if it feels good, I'll keep exploring the math consistently, which is really the foundation of building skill.

Consistent effort through time.

Learning the shortcuts, how to be more efficient, all of the nuances will come naturally so long as I spend enough time exploring consistently.

So keep that in mind, don't worry about time or getting the "right" answer (whatever that means).

Become a detective to find an interesting path to YOU, and then you will fall in love with math, and you will unlock an entire universe of new possibilities because before you know it, you'll be speaking and even thinking in the language the universe is written in.