The Essence of Mathematics Through Elementary Problems

If Lana was 31 years younger, she'd be 12 times as old as her baby. Lana is half as young as Robert, who is 44. Find the baby.

Which one was the best you read?

I would expect that quite a few determined children could do so. The immediate next question is "first ten powers of 2" and not to gatekeep but surely vast swathes of the commentariat here knew most of those pretty young.

Quickly running through it I know from memory all but a couple of cubes, all but maybe ten squares, all of the multiplication tables. It would take maybe 30 minutes to get everything down

I'm pretty handy with numbers but nothing special. It doesn't seem like that major a task

We had to memorize the squares up to 21 in grade school.

The book is absolutely right though -- memorizing squares/cubes/2^n is a math superpower. It's extremely useful when solving textbook problems or when just exploring math on your own. I've seen the powers of 3 so often they've immediately become recognizable, 27,81,243, just as I'm sure most people immediately recognize 81 as a square when they see it -- it just "pops out" from the page.

This holds for simple identities too. Having things like x^2 - y^2 = (x+y)(x-y) immediately come to mind when you see just one half of the identity is ridiculously useful.

Note that the first problem is not actually to learn the multiplication tables! Problem 1 is to calculate a few things like 0.004×0.02 and 1.08÷1.2 in one's head; it's just that these real questions are placed in 1(b), with a precursor 1(a), purely to indicate prerequisites, that states "Compute for yourself, and learn by heart, the times tables up to 9×9." (Incidentally, even here the "compute for yourself" part is the real heart of the matter, though of course knowing the times table up to 9×9 is something very useful. You'll see a lot more connections once there's some "fuel", as mentioned in the preface and the introduction to the chapter.)

In any case, don't let a cursory look at Problem 1 decide the nature of the book for you. There's more than meets the eye. I don't know much about Tony Gardiner but the other author, Alexandre Borovik, has a wonderful blog called "Mathematics under the Microscope" (https://micromath.wordpress.com/) (and also a book of that name); to everything he brings a profound perspective with a light touch, a unique combination. A couple of memorable posts I still remember from over a decade ago:

• “Psychophysiology of blackboard teaching” https://micromath.wordpress.com/2010/09/24/psychophysiology-...

• “Achilles, Tortoise and Yessenin-Volpin” https://micromath.wordpress.com/2009/02/16/achilles-tortoise...

• All the posts about "a childhood story".

Anyway, I haven't gone through this book so far, but skimming through some chapters and how they are put together, there is a lot of gold here. See for instance Problem 78, which is simply twenty opportunities to think about "3 - 1 = 2".

Mathematician's Delight by W. W. Sawyer seems a good book.

I like to point out to people that to learn the times tables up to 10, there are only really 36 facts to learn, since anything times 1 or 10 is trivial, and multiplication is commutative. (That's already a nice little problem to look at, to show that it is 36.)

And then there are many interesting tricks to approach those 36. For example, the 'fingers' trick for the 9 times table. (Which leads to consideration of why do the digits of a multiple of 9 always add up to a multiple of 9?) Muliplying by 5 is half of multiplying by 10. Multiplying by 4 is doubling twice, etc.

Before you know it, you just have to drill the ones most common to choke on (like 7x7, 7x8).

Each chapter ends with a "Comments and Solutions" in smaller print; as far as I can tell every single problem has solutions, sometimes multiple pages' worth.

From the copyright notice inside the opening pages:

> 2019 Alexandre Borovik and Tony Gardiner This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the work; to adapt the work and to make commercial use of the work providing attribution is made to the author (but not in any way that suggests that they endorse you or your use of the work).