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



How to Manage Your Narrow and Broad Focus Attention (4/4)

As we discussed in our previous post, our voluntary attention comes in two flavors: broad and narrow. Broad focus attention is great for getting your bearings, understanding the “big picture,” and comprehending complex systems and relationships. Narrow focus attention allows us to be efficient, productive, and meticulous.

To be effective supreme commanders of the mind, we need to know when to use one and when to use the other; sometimes you want to be holed up in your war room, poring over plans and maps, and sometimes you need to go out to the frontlines to see exactly what is going on on the ground.

Knowing when to use a broad or narrow focus attention is more art than science – it’s something you have to learn from experience; however, there’s actually a science to how you shift into those different attentional foci. Below we provide a few researched-backed tips:

Narrow Your Focus

Use lists, outlines, and categories. When we categorize, use lists, or create outlines, our attention narrows in order to pinpoint any missing information. If you’re working on a task in which getting details right is vital, write out all the steps or even use a checklist.

Focus on a goal. The fact that having a clear goal can narrow one’s focus is perfectly displayed in the Invisible Gorilla Experiment. When the experiment’s participants were told to watch a video and given a goal to count how many times a basketball was passed around, they became so narrowly focused on the ball that they failed to see a man dressed in a gorilla suit stroll casually among the players and dance in the middle of the court.

While a goal is an effective attention narrower, there’s a risk of suffering tunnel vision and missing out on more rewarding opportunities. Always employ your practical wisdom.

Take it slow. When you think, read, or observe your surroundings slowly, your attention narrows. You’ll spend more time homing in on and examining the objects in your environment that catch your involuntary attention and use your voluntary attention to ponder and analyze single words and sentences within a large piece of literature.

Broaden Your Focus

Stay optimistic. Research has shown that positive emotions give us a more open attention. When we’re optimistic, we’re relaxed and thus more likely to see new connections and opportunities. This is one reason why it’s so important that leaders remain upbeat; a sense of realistic optimism is essential in crafting and maintaining a strategic big-picture vision.

Focus on others. Another way to broaden your attention is to shift your focus from yourself and onto others. Studies show that being “other directed” or thinking in terms of “we” and not “me” opens up attention. The best way to make that shift is to simply help another person with a problem. You can also try doing some “compassionate meditation.”

Scan. When we quickly scan our environment (or even a book), our attention widens in order to take in as much information as possible, which in turn allows us to get a quick and dirty overview of the situation or text.

Gather contrary evidence. Once we decide that someone has an inherent flaw and we label them with it – they’re stupid, crazy, useless, selfish, immature, bitchy, evil, lazy, etc. – a narrow focus tends to set in. You experience the Velcro/Teflon effect: you notice everything the person does that confirms your conclusion, but overlook any conflicting evidence.

If you find yourself only being able to see a loved one through the lens of a negative label, it can help to actively look for things they do that run contrary to it, and even write those things down. While lists can narrow your focus in some cases, they can also be used to produce a broader, more balanced picture in others. Think for example of keeping a gratitude journal; if you find yourself narrowly attuned to what’s wrong with your life, making a list of the good things can greatly broaden your perspective.


If you want to win the war on distraction and build an empire of personal progress, you need to be a wise supreme commander that knows how to best utilize his units. Sometimes you want to send one type of your attention to the frontlines, and sometimes you want to send another to the rear for rest. By deftly maneuvering your resources and effectively deploying your troops, you can make the most of your invaluable attention.

Of course the pure strength of your fighting force matters greatly too. Single-minded focus may be only one element of your attention, but it’s still vital one. But since this article has been so long and meaty, and your voluntary attention is now all tuckered out, it’s time to let it get a hot meal and a shower. For instructions on how to strengthen your powers of concentration, return to the briefing room tomorrow at 1900 hours.

Mastering your attention 3/4 (How to Manage Your Mind Wandering)

Sometimes your voluntary and involuntary attentional modes need some R&R, and the cognitive equivalent of the USO is a good old-fashioned mind wandering session.

While mind wandering (or daydreaming) can boost creativity and help us untangle unresolved problems, it can also distract us at inopportune times and lead us to ruminate on negative thoughts and emotions. Thus while daydreaming may seem the ultimate in creative spontaneity, to maximize its benefits and minimize its drawbacks, it’s best to actively manage your mind wandering sessions:

Intentionally set a time to let your mind wander. Instead of limiting your daydreaming to those few abbreviated pockets between when your mind unintentionally drifts away from the task at hand and when you yank it back to work, find times throughout your day where you deliberately give your brain permission to wander at will. Some great thinkers and leaders have made it a habit to block out chunks in their day where they don’t do anything except let their mind freely ramble.

Besides blocking off specific time in your schedule for mind wandering, give your brain permission to wander when you’re doing low-cognition activities like cleaning, whittling, or showering. A bit of habitual stimulation really seems to free the mind up to receive inspiration. If you find yourself stuck on a problem, instead of sitting there trying to force the solution from your cranium, take a break and the answer may very well come to you in the shower.

Decide what kind of mind wandering session you want to have. When we daydream, our mind has a tendency to drift towards negative thoughts and emotions. It does this in order to direct our attention to unresolved problems in our lives. This can be beneficial, so it’s good to intentionally set aside times when you give yourself permission to be a worrywart. Make a list of everything that you’re worried about. Next to each worry, write down a “next step” – something tangible you can do, however small, to begin resolving that issue. If there’s truly nothing you can do about something for the time being, make a conscious note of that and imagine tabling the issue for another session.

Sometimes though, we don’t want our cognitive rambles to drift over to the dark side and be such a downer. Instead, we’re hoping our daydreams can generate a bit of inspiration or creativity. In that case, actively focus on positive thoughts as your mind wanders. If it starts to drift towards more negative things, nudge it back on course. It may help to keep a mental drawer of positive subject file folders you can leaf through – fond childhood memories, things you love about your girlfriend, the last vacation you took, and so on.