Numberplay: Ordering in Circles - NYTimes.com Take a look at the numbers in the inner black circle above. This clockwise arrangement of 1, 2, 3, 1, 3, and 4 has an interesting property. With it you can count all the numbers from one to seven in order in the following way. You go around the circle reading each number and if it is in counting sequence starting from one, you count it and move on. If it is not in counting sequence, you are allowed to add as many consecutive numbers as needed to make it so. You can follow the process on the outer blue circles above. The first three numbers on the black circle are 1, 2 and 3. They are in counting order, so you merely count each one and move on. Next you encounter a 1 so you add the next number which is a 3 to get four. Next, wrapping around the circle, are 4 and 1, giving five. The next three (2, 3 and 1) add up to six, and the next two (3 and 4) add up to seven. This very simple process hardly seems to have anything that can cause bafflement. Yet, like the somewhat similar de Bruijn sequence that we saw in The Math Behind the Magic, this order-generating sequences of small numbers forms the the basis of some very clever puzzles served to us by Dave Fergemann, one of most creative puzzle solvers here at Numberplay. Dave had previously given us a puzzle that linked music, house building and presidents: Building Walls Rationally. As Dave points out, the pattern in the circle above breaks down after seven: there is no way to make a total of eight starting from the first 1, the place where we reach after counting seven. Dave continues: Now, if the six numbers had all been 1, we could have kept counting forever by grouping as many ones as we needed at each step. This is a trivial case, though. What makes the example above more surprising and interesting is that most of the numbers are larger than one, and yet they all seem to fall in place to give us the right totals as we count. So in the following questions let's try to use the largest numbers possible, as measured by the the sum of all the numbers in the circles. See if you can find six non-trivial numbers that allow you to continue counting up to at least 15 in the manner described above, while keeping the sum of the numbers as large as possible. What is the six-number sequence with the largest sum that allows you to count up to at least 15? Using your answer to question 1, how much higher can you continue to count? Either find the first point where the pattern breaks down, or prove that it never does. If we want an even larger sum, we'll need more numbers around the circle. Aiming for a sum of 21, what's the shortest sequence of numbers that allows you to continue counting forever? What about for a sum of 45? Can you come up with a simple method for finding the shortest such sequence with a given sum? This question is more trivia than puzzle. The sequence of numbers that best answers question 1 had a particular technological application at least as far back as the 1400s. I first learned about it, as well as this sequence, from an exhibit in the British Museum. There is still a working example of it at a historic landmark in Prague. Can you identify this invention? And here are two word challenges, involving circles of letters instead of numbers. If you write the word demo in a circle, then you can make a word starting from any one of the four letters: demo, em, mode, and ode. Can you find other words with the same property? We'll score these based on the total length of all the words formed. The score for "demo" would be 4 + 2 + 4 + 3 = 13. How much better can you do? If you write the letters nototitis in a circle, you can follow the circle twice around to reveal a doctor's assessment of a young patient's earache: "No, tot, it is not otitis." Can you come up with other circular sentences? Try to go at least twice around the circle, but divide up the letters in a different way each time. "A rose is a rose is a rose" doesn't count! Note that an answer to question 5 must be an actual word, but answers to question 6 do not have to be. A hearty thank you to Dave for this interesting puzzle made from such a simple idea. Though it deals with simple operations on simple numbers, this set of puzzles captures the heart of creative mathematical thought as we've often seen before, as for example in The Circle of Differences. You play with simple specific cases and suddenly find that there is something deeper - and if you're lucky, you have an inductive insight that reveals this deeper regularity. That's the joy of Numberplay.