Wrapping a Cube

This puzzle comes from a wonderful Russian site, where its solution is presented as a sequence of animations. (A later remark: through the efforts of Colm Mulcahy who approached David Singmaster an earlier reference has been found: Martin Gardner describes the puzzle at the end of Chapter 5 of his New Mathematical Diversions – a book from 1995. Colm Mulcahy later reported that the article was included into Gardner’s 1966 Third book of Mathematical Diversions from Scientific American and later added that the original column appeared in 1960.))


Is it possible to wrap the cube with a 3×3 piece of paper below it?

Handling of the paper is subject to two conditions:

  1. The paper may be only cut or folded along the crease lines.
  2. The cutting should not cause pieces to separate.

Would you like to check? Here is the answer



It’s all your fault

You know those people who always seem to blame their misfortune on everyone but themselves? I know that I know them. It’s always his fault or her fault or their fault or the world’s fault. It’s even just plain bad luck or things just never seeming to go their way. It’s always someone or something else’s fault. It’s never their own. As if everything bad that happens in their life is part of some well orchestrated grand conspiracy.

And even if this is not who we generally are, we all have these singular moments. Moments where we are quick to point fingers and assign blame. Moments when, if something does not go as well as expected, it is not our fault — it’s theirs.

My first thought when I encounter such people or situations is this: Why would anyone give all of their power away so easily?


Yes. Power.

You see, if you are of the mind that everything bad that happens is someone else’s fault, or if you think that life just kind of happens to you as you are living it, then you are assigning a tremendous amount of power to them and assume no power to be able to change it yourself. If your choices and actions are always a reaction to the things that they do, then you have no agency to take action or make the independent choices that drive your life. And, I can’t imagine a life more sad than one where you believe that bad things just happen to you for no good reason. Where others are mean or things don’t go your way and you have no ability to make it stop.

But, if you see things differently and assume the responsibility for the things that happen in your life, then you also command the power to change them. Once you stop believing that them or they or the man or life are to blame, and start to look within for reasons instead, you can start about the work of making the changes needed to turn the tide.

The position of power is the ability to accept responsibility for one’s fortune — good or bad. Because only in this position does one have the power and opportunity to change or sutain it.


About being sorry

Apologies, true ones, are more than these two words. Apologies require action. In my view, there are at least four separate steps required for any apology to be considered valid:

Recognize what you did wrong, why, how, and to whom.
Apologize to the person/people you wronged. Ideally with an explanation, now that you have completed step one and can provide a true one.
Remedy the wrong in whatever way possible. Preferably in a way that those you’ve wronged see as proper remedy as well. If they suggest one, and it is in any way agreeable, by all means save yourself some trouble and do that.
Learn from the experience so that you may avoid doing such wrong again. This is especially important because doing such again not only effectively cancels out the previous three steps but likely may render another round of apology ineffective or the requirements needed for another acceptance insurmountable.
Two simple words are never enough. In fact, they are all too often not even the right place to start.

Lately, I’ve been thinking that most of us could use a remedial course in such a basic courtesy. Heck, I’ll be the first to admit that I’m just as guilty and in need. So consider this simple message a note to self that I hope others can learn from too.