I showed my small son Sam the Bridges of Königsberg problem:

The problem is to find a walk through the city that crosses each bridge once and only once. Turns out that this is impossible: I told Sam, but he kept trying anyhow. He ended up pretty frustrated. Later, over supper, I told him that Königsberg was now a Russian city called Kaliningrad – that it had been captured in WWII, in a vast assault involving thousands of tanks (3rd Belorussian Front). Sam, assuming the Soviet motives were the same as his, pointed out that they only really needed to have blown up one bridge.

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There is another solution that is basically complementary to your son’s.

This problem was supposedly offered to Kaiser Wilhelm II at a social event. He claimed that he could solve it in less than two minutes. The person who proposed the problem was amused, knowing full well that the problem was unsolvable, but decided to see what would happen. Kaiser took a piece of paper and wrote an order to build an eighth bridge over the south fork of the river.

The bridge was in fact built sometime around 1905 and became known as “Kaiser’s bridge”.

Then in 1942 the whole city was extensively bombed, taking down two bridges and bringing the number down to 6.

That is adorable.

Here’s what it looks like now:

https://maps.google.com/maps?hl=en&ll=54.702953,20.517955&spn=0.011531,0.030556&t=h&z=15

There are eight bridges connecting the two islands and the problem is easily solvable. The two bridges on the right (east) are under construction in the Google maps photo but were finished last summer.

I prefer Euler’s solution: invent topology.

Unrelated, but in case you have not read the Arxiv pre-print for “The genetic architecture of adaptations to high altitude in Ethiopia,” here is a section that will likely be of some interest, as unsurprising as it is: “With regard to O2 sat, HA Amhara had a 5.6% lower O2 sat compared to LA Amhara while HA Oromo had 10.5% lower O2 sat than their LA counterparts. Therefore, we detected significant phenotypic differences not only between populations from the same ethnic group that live at different altitudes, but also across populations from closely related ethnic groups (Oromo and Amhara) that live at the same altitude. Given the low genetic divergence between these two ethnic groups at the genome-wide level (mean FST = 0.0098), the phenotypic differences between Amhara and Oromo highlanders are unlikely to be due to independent genetic adaptations in these ethnic groups; rather they are likely to reflect genetic adaptations that evolved in the Amhara, due to their longer residence at HA.”

Consider this data point number 10,000 that the important thing is not genome-wide FST, but rather directional selection acting on correlated loci to produce phenotypic divergence.

Cochran, totally off-topic, but I saw this paper and thought of the ‘new germ theory’

The Fox and the Rabbits—Environmental Variables and Population Genetics (1) Replication Problems in Association Studies and the Untapped Power of GWAS (2) Vitamin A Deficiency, Herpes Simplex Reactivation and Other Causes of Alzheimer’s Disease

When I was 10 years old, but already quite ADHD-OCD-posessed by mathematics, I wasted lots of time trying to solve this one in my school notebooks. (I also wasted time a little later on trying to find an algebraic expression for the sine of 1 degree – also impossible.) The only cure seems to be teaching the afflicted one the relevant mathematics. In this case, it’s just graph theory. Best,

Jim Given

Speaking of, i was just reading before coming here:

“The German weekly Der Spiegel published the sensational details of a little-known offer by the Soviet leadership to return Kaliningrad, the former Königsberg, to Germany in 1990.”

“The Germans, however, were lukewarm about the offer, allegedly telling the Soviets that reunification was the country’s priority and that it regarded the Kaliningrad region as a Soviet problem.”

from The Guardian: Should Kant’s home once again be German?

See my post 2 down.

Tom Stoppard’s play “Hapgood” has the plot hinge on another solution: You can do it if you have two walkers. Betweed that and “Arcadia” – and hell, Rosencrantz, Jumpers, and a few others, come to think of it – Stoppard has picked up a fair bit of math and made stories of it. Quite good for a dropout, eh? Almost makes one think that something other than environment might be in play for cleverness.

Did it include a “german corridor” to connect Kalingrad (east East Prussia) to Germany considering that the German border moved several hundred miles to the west around 1945?