News and Notes

(61012) Another very old problem (Challenge Problem
11 from 2003) has finally had its solution posted.

(81511) The solution to the (very) old backlog problem
Advanced Problem 98 has been posted. More to come (one hopes).

(80311) The solution to the number of resonance structures of buckminsterfullerene
was posted (Challenge Problem 4, 1011), but an analytic
(rather than computer) solution is still sought.

(62711) The solution to the problem of placing 24 circles so that each is tangent to
exactly three others (Challenge Problem 1, 1011) has
been posted. There it is shown that no such configuration exists for an odd number of
circles and that any even number of circles greater than 14 can be realized. The status
of small even values remains to be determined. The question of how many combinatorically
distinct configurations of 24 circles is also open. See the
solution.

(73010) The dissection problem posed in Challenge Problem 6 from 0506 was done with a minimal number of pieces by James H. Caldwell in 1964. See the solution.

(72810) The solution to Challenge Problem 4 from 2004 has finally been posted.

(72810) Challenge Problem 9 from 0506 is more difficult than I'd anticipated. See Lights Out Puzzle at MathWorld.

(63010) Bojan Basic has found all 18 different ways of cutting a triangle with three cuts (Challenge Problem 2 from 20092010) and provided a lower bound of 136 (which he believes is in fact the answer) ways of cutting a triangle with four cuts. His solution is here. Can anyone prove that the answer is, in fact, 136?

(60110) No solutions have been submitted to Challenge Problem 2 (update: A partial solution has been given. See 63010.) or Challenge Problem 3 from 20092010.

(51810) In an unpublished note (http://arxiv.org/pdf/0804.1307)
S. Kurz and A. Wassermann claim to have the exact solutions to Problem 9 from 0809 up to n = 36.

(20710) Bojan Basic points out that Problem 9 from 0809 is an open research problem.
Bojan Basic's remarks and a partial solution (and algorithm) due to Christian Bau are
here.

(81909) The solution to Challenge Problem 6 from 0809 has been posted. The problem
of determining all configurations of five points in space such that the set of pairwise
distatnces between them has exactly two elements was more elaborate than I thought
(there are 27 of them if mirror images are considered equivalent). I hope to eventually
post images of all of the configurations.

(6709) Jonathan Welton gives examples of disconnected "knight's graphs" on 4×12
and 4×16 boards that outperform connected ones.
Kirk Bresniker gives such an example on the 7×7 board.
(Challenge Problem 12 from 0708; see the
solution page)

(52409) Erich Friedman reports on Jonathan Welton's progress on
Challenge Problem 12 from 0708. See the
solution page.
 (52109) Problem status has been recategorized:
* Denotes a problem which has not yet been solved.
× Denotes a partially solved or openended problem.
o Denotes a work in progress.
+ Denotes a problem whose solution has recently been posted.
 (21609) The solution to
Challenge Problem 2 from 0708 has been posted.
 (2309) Dave Beckman notes that the solution to
High School Problem 114 was missing two triangles.
 (2309) Dave Beckman has found other prime factors of the number
20001999...4321 introducted in Challenge Problem 4
from 9900. His results have been added to the solution.
 (12709) Jim Boyce has solved Challenge Problem 1
from 0809.
 (12708) Claudio Baiocchi has given a construction to produce maximal solutions
to the generalization of Challenge Problem 11 from 0708.
See the solution page. Using his construction, the
maximum for an n×n array is
n^{4} − 4n^{2} + 8n − 11 for n > 1, n odd
n^{4} − 4n^{2} + 8n − 8 for n even.
It would be nice to see what the answers are for m×n arrays and
n×n×n arrays.
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This page is maintained by Les Reid. Last updated 62711.