»  Solutions to puzzles in my VDARE.com monthly Diary

  January 2021

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In the Math Corner of my January Diary I offered the following brainteaser …

The Vukmirovic Sequence:  Let x1, x2, and x3 be real numbers, and define xn for n ≥ 4 by  xn =  max{xn−3xn−1} − xn−2. Show that the sequence x1x2, … is either convergent or eventually periodic, and find all triples (x1x2x3) for which it is converegent.

This was, I explained, one of the problems — problem number 12226 — in the January issue of MAA Monthly, submitted by Jovan Vukmirovic of Belgrade, Serbia. I hereby christen it the problem of the Vukmirovic Sequence.

As I further explained, there is a lag of about fifteen months between the journal posting a problem and posting a worked solution to the problem. So a worked solution to this one will be posted sometime in Spring of 2022. In the meantime we must do the best we can.

My best has not been very good. I have some notes in the February Diary. Readers have been no more successful. We have plenty of particular results, but no general proof. Work continues …

[Added October 2022:  The solution is here.]