I'm on the road this week. Save for a couple of brief layovers, I've been doing a lot of driving. In light of this, I was inspired to feature a puzzle that could be solved without pen and paper. It was while glancing at my hands (which were, at the time, wrapped around a steering wheel) that this classic brain teaser came to mind.
Photo Credit: Nikola via flickr | CC BY-NC-ND 2.0
What is the highest number you can reach by counting with the fingers on both hands? Assume, for the purposes of this question, that thumbs are fingers, and that you are finger-counting with a total of ten digits. (Once solved, it should be straightforward to determine maximum countable values for N-finger scenarios.)
We'll be back next week with the solution – and a new puzzle! Got a great brainteaser, original or otherwise, that you'd like to see featured? E-mail me with your recommendations. (Be sure to include "Sunday Puzzle" in the subject line.)
Last week I asked you to turn four three-link chains into a closed loop of twelve links, and to do so by cutting a re-welding as few links as possible.
The solution to this puzzle is short and sweet. It's three. Three is the minimum number of links you must cut and re-weld to create the circle of twelve. If you cut each of the links in a single three-link chain, you will wind up with three open links that can be used to join the remaining three-link chains together. Joining the three three-link chains (9) with three individual links (1+1+1) gives us the closed, twelve-link loop.
The key to arriving at this answer is to avoid becoming too wedded to the idea that closing the loops requires you to modify each of the four three-link chains individually; what makes this puzzle challenging, I think, is that most people (myself included) start down this path right away, noticing, for example, that cutting the last link in each individual three-link chain allows you to re-weld the segments together nose-to-end, closing the loop with four cuts and four re-welds. But once you head down this road, it can be difficult to reverse course. If our sole approach to this problem is to grant each three-link segment unique attention, we will never give our brains the space required to approach the puzzle from a new angle. And in the absence of fresh perspective, the puzzle goes unsolved.
You're going to find yourself on the wrong path a lot. That's okay. The fact is that taking an incorrect approach, while undeniably frustrating, is one of the most common and universal aspects of the problem-solving process. A more problematic pitfall, by far, is being unwilling or unable to abandon a failed approach. Doing so is rarely easy, but it gets less difficult with practice. The same can be said for two other, related skills: the ability to set about a puzzle with fresh eyes, and the ability to accurately sense when you're on the right path.
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