Kipli's Cage

Daniel Horsley, a doctoral student at the University of Queensland, Australia, has reportedly solved Lindner's conjecture on Steiner triple systems.

Let `V` be a set of size `v`. A Steiner triple system (of degree `v`) is a collection of subsets of size 3 such that each pair of elements in `V` appears exactly once in an element of the collection. Put another way, a Steiner triple system is a partition of the edges of the complete graph on `v` vertices into triangles. For example, a Steiner triple system of degree 7 is illustrated below.

There are Steiner triple systems of degree `v` if and only if `v` is congruent to 1 or 3 mod 6. The number of nonisomorphic Steiner triple systems increases very quickly: at `v = 13` there are 80 systems; at `v = 19`, there are 11084874829 different systems.

Lindner's conjecture is about embedding partial Steiner triple systems (a partition of some subset of the edges of the complete graph into triangles). Lindner conjectured that any Steiner triple system of degree `v` can be embeddedin (extended to) any Steiner triple system of degree at least `2v + 1` (and also congruent to 1 or 3 modulo 6).

Steiner triple systems show up in combinatorial design theory, an area of mathematics that aids in the design of statistical experiments (though, as is usual, many important mathematical concepts predate their application and go beyond the applications today). For example, given a collection of different varieties of seeds that we would like to test for (say) hardiness, and a collection of plots where we might plant the seeds, how do we assign seeds to plots so that we can discern differences in seed varieties from differences in land plots?

More information on Steiner triple systems can be found at mathworld and on design theory at DesignTheory.org and the DMOZ Open Directory entry on Design Theory.

The 193-digit number

310 7418240490 0437213507 5003588856 7930037346 0228427275 4572016194 8823206440 5180815045 5634682967 1723286782 4379162728 3803341547 1073108501 9195485290 0733772482 2783525742 3864540146 9173660247 7652346609

has been factored. Mathworld explains why that is a big deal.

This is curious:

11 Jordanian officials resign after bombings

AMMAN, Jordan (AP) — Eleven top Jordanian officials, including the kingdom's national security adviser, resigned Tuesday in the wake of last week's triple hotel bombings, state-run TV announced.

[snip]

The TV announcement made no reference to last Wednesday's attacks on three Amman hotels that killed 61 people and dented the reputation of the country's revered security services.

But the bombings sparked national outrage and raised concerns over the handling of the security services.

There has also been criticism over how the four Iraqi cell members entered Jordan on November 5 without being detected before carrying out their attacks.

Compare what happened to these Jordanian officials with what happened in the U.S. after 9/11. Were there any resignations of top officials here? Did anyone really accept responsibility? Was anyone held accountable for security failings?

Now, I do remember that Bush chastised George Tenet, the director of the CIA at the time. I fact, here's a video of it and even a nice closeup shot of a chastised Tenet. Oh...wait a sec...he wasn't being chastised was he? No, Tenet was given a freakin' Medal of Freedom!

Grrrr....

The attack in Jordan killed over 60 people, which is certainly horrific. The WTC attack killed 3000. The attack in Jordan was simple and required relatively little planning. The WTC attack involved coordination and planning that stretched over many months, if not a few years. The attack in Jordan was carried out by people who needed to travel only a short distance (from Iraq) and who didn't need to stay 'in-country' for an extended period of time. The attack on the WTC involved people who lived in the U.S. for years and some flew regularly in and out of the country.

If any officials needed to resign, it was those in the U.S.

Show that there are no five-digit numbers such that if you remove the first two digits and place them at the end (e.g., 12345 to 34512) then the resulting number is half the original.

Note: of course the number 00000 would work, but we will ignore that trivial case.