![]() Spider Sequences: Find the next term of the given number sequences. Sign Sequences: Continue the sequences if you can work out the rule. Sequence Dancing: Find the next term of the number sequences. Sea Shells: A question which can be best answered by using algebra. One one: Continue the given number pattern with the help of a little lateral thinking. Missing Terms: Find the missing terms from these linear sequences. House Numbers: The numbers on five houses next to each other add up to 70. Let a(k+1) = x + h and d(k+1) = h, and do Step 1.Add 'em: Add up a sequence of consecutive numbers. Let h be the least positive integer not in D(k) such that x + h is not in P(k). If there is a negative integer h not in D(k) such that x + h is not in P(k), and x > 0, then let d(k+1) be the greatest such h, let a(k+1) = x + h, and return to Step 1 otherwise do Step 2. Let a 1 = 1, and for n > 1, define a n = if this number is not in the set. SOLVED by Mateusz Kwasnicki, January 2004 The problem is to prove or disprove that every positive integer will eventually be written during the counting procedure. Where all the a(i) and b(i) are positive integers, and the b(i) are distinct. ![]() It's just like the special, except that we start with an arbitrary initial counting that is, instead of starting with one 1, start with Now for the general form of this problem. If this procedure continues indefinitely, will every positive integer eventually be written? Skip some space, and count what you've written so far by writing Skip some space, and count the four 1's and one 3 that you've written so far by writing Skip some more space and count the three 1's you've written so far by writing "3 1" - except write it with 3 above 1, like this: Skip some space and count the one 1 you've written by writing "1 1". Kimberling, Problem 2386, Crux Mathematicorum 24 (1998) 426 and is equivalent to the following: Like the theory of relativity, this problem has a special version and a general version. The solution appears in Crux Mathematicorum 29 (2003) 320-321. Any binary sequence can serve as a choice-sequence details are given in the reference cited just above. The word R corresponds to a choice-sequence consisting of all 0's. The problem has been solved in the affirmative: the binary word R does indeed contain every binary word infinitely many times. Kimberling, Problem 2289, Crux Mathematicorum 23 (1997) 501. This problem originates, in more general form, inĬ. If so, then it is easy to see that every word repeats infinitely many times in R, which is notable, since the rule for generating R tries to resist repetition. Has greater maximal repeated segment length thanĪ finite string of 0's and 1's is called a word. The sequence R is constructed to avoid repetitions "as long as possible", as follows: The first segment of length 3 to repeat is "010", at r 10. The first segment of length 2 to repeat "00", at r 5. The first segment of length 1 to repeat is "0", at r 1 and r 3. Kimberling, Problem 1615, Crux Mathematicorum 17 (1991) 44. Guy, Unsolved Problems in Number Theory, second edition, Springer-Verlag, 1994.Ĭ. To generate the sequence, visitįor a discussion and variant of the problem, see See also Kolakoski Sequence at MathWorld and Sequence A000002 at the Online Encyclopedia of Integer Sequences. It was discussed by Rufus Oldenburger, "Exponent trajectories in symbolic dynamics", Transactions of the American Mathematical Society 46 (1939), 453-466. However, it quite clearly occured earlier and can now be called the Oldenburger-Kolakoski sequence. For a proof that the Kolakoski sequence is not periodic, see the same Monthly 73 (1966) 681-682. William Kolakoski, "Self generating runs, Problem 5304," American Mathematical Monthly 72 (1965) 674. Reward: $200.00 for publishing a solution of any one of the five problems stated in Integer Sequences and Arrays.įor many years, the sequence was thought to originate as indicated here: This sequence is identical to its own runlength sequence. If you are first to publish a solution, let me know, and collect your reward! Or, if you find a short solution and you are quite sure it is correct and complete, send it to If accepted, your proof will be published on this site - see, for example, Problem 8.ġ. Stated below are a few challenging problems. UNSOLVED PROBLEMS AND REWARDS Unsolved Problems and Rewards
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