Posted by: Alexandre Borovik | April 18, 2010

An anonymous response to the “buzzle”

Dear Alexandre,

the comments regarding your beautiful “buzzle” are not very enlightening.

My thought process upon seing the bus:

1. I analyzed the picture: where is the clue, where is the difference? I found out that there is no visible clue. I made shure that there is no visible clue checking details and symmtery.
Both halfs of the bus ar identical, no hint outside the bus to the left or the right.

2. Either this is something really clever, or it is a trick being played on me. (With great psychological exegesis regarding the choice of left or right – usually of the kind: see what tricks your own brain is playing on you and how stupid and unconscious you are.)

3. I opt for really clever, because I like to solve. It is like searching for easter eggs. What a joy when you find one. In case it is “psychology” – that can come later.

4. If the clue is not in the visible I have to look for the invisible.
If children can solve this, what do they do?

5. I start moving the bus in my imagination from right to left and from left to right. I start to imagine children in the bus waving at the windows. I hear them laugh and chat and all the chaos in the bus. I start to imagine being a pre-schooler going with the bus with my pals. I am there with them.

6. Bingo! We are stopping at the next station, the doors open and other friends come in. Where are the doors? Left, left, left, left! The bus is driving to the left. So simple!

7. My adult brain checks again. Is it correct that the doors are allways to the right of the driver? Yes. The bus is allways driving this way, stopping this way: Hello children. Closing the doors again continuing the drive.

8. Children in the UK, Ireland, in India and in Australia will answer to the right. The driver is sitting on the right side, the bus travels on the left lane, the doors are on the left. (See

This is not proto-mathematics. This is higher mathematics. You just called it proto-mathematics because you couldn’t solve it with your “elementary school” mathematics.

Some reflections:

Children are not stupid. Children are wide open and they think inclusively vs. separatingly.  The thought process is still a different one.

The newborn is radically different than we are now. Inner and outer, I and you have not yet cristallized and stabilized. This is a long process taking the first 3 to 5 years at least.

The world is a mystery and a miracle. So many perceptions coming through the senses, the emotions and thought process. In the struggle to orient yourself and get in charge, learning to control your body and fulfilling your needs and desires you start organizing your perceptions (the world) and you start realizing things like before and after, yesterday, now, tomorrow (time), inner and outer, me and you (body, space) and then one day after having worked with it for a long time already the full implication of the general principle of causation dawns upon you: given certain circumstances when A then B. Allways. Reproducible. Doable. Usable. Predictable. What a powerful thing.

In order to make theese discoveries all perceptions have to be connected and related one to another. As long as I cannot isolate certain perceptions in order to explain with sufficient certainty a cause and effect relationship between A and B I am dealing with thousands of “accidental” concomitant obeservations. So my mind is wide open and paying attention to all there is registering “it” in its totality and completeness as far as my senses, capability of processing the input and my memory allow.

Once I found out about a causal relationship I can safely limit my observation to the necessary elements regarding this special relationship. If I want to open the fridge and get at this or that, I do not have to pay attention to the sound of the fridge (is the fridge friendly now, or will it bite me?), to the reflections of the light on the fridge (is the fridge playing with the sun now, can I molest it, or will it get angry?), etc. I know where and how to open it and I “know” that the fridge is a dead machine. (Is it?)

Growing up in an adult world, we learn to limit and limit and limit ever more, until we become so stupid, that we cannot solve a pre-schooler buzzle.

The next stupidity we tend to commit is juxtaposing the modes of perception and world-construction trying to figure out which one is better. It allways has to be either this or that.
Either the children’s mode or the adult’s mode. How limiting again.

The point to be made is that we have to learn to become as the children (isn’t that the hint the famous sourcerer of the bible is giving us: you have to become like the children in order to enter the heavens?). Once we have mastered the causal path and learnt to cristallize and and limit we have to learn how to undo again and how to move from one direction into the other and back again. Like breathing – in and out.

And then we should learn how not to abuse and deform our children to loose that precious mode of being during the process of learning the other mode, the limiting, separating, causal mode (which is too is a most precious mode of course).

Learning new things means the ability to dissolve again in order to produce new cristallizations. Solve et coagula – the motto of the alchemist. It is a mind process as well as a chemical process.

Neither the dissolvers nor the cristallizers will save the world. The alchemists are our saviours. Lets get on the bus and have a joyful ride with our children.



  1. This puzzle may be interesting, but I don’t believe any preschoolers came up with the solution.

  2. This is not proto-mathematics. This is higher mathematics.

    Really? Someone’s really high, it seems!

  3. This is mathematics-as-puzzles-and-tricks – the solution is found by ad hoc reasoning, not applicable to any other problem or domain, and not revealing of any deeper mathematical or inferential structure. What exactly have you learnt if you you learn the solution? Nothing apart from the buses are not laterally symmetric! Is that useful knowledge? Perhaps for bus-drivers and their passengers, but certainly not for mathematicians.

    If you like puzzles and tricks and games and one-off solutions, then this is the stuff for you!

    If however, you are interested in deep structures, in deductive consequences, in methods of reasoning or abstract theories which you can be apply to other problems or other domains, then this is not mathematics.

    It is a great shame that our children should be given such nonsense under the name of mathematics!

  4. I got the anwser straight away. Assuming the bus isn’t going backwards then it would always be moving right. I don’t see what hard about the question.

    Note, I live in the uk.

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