# Chaitanya's Random Pages

## July 31, 2010

### Tough Tour de France climbs

Filed under: sport — ckrao @ 12:25 pm

Every year I marvel at the level of fitness by the Tour de France cyclists in not just riding, but racing, up extraordinary heights. The toughest mountain stage of a tour may have 3 climbs each with over 1000m elevation gain! Here are some of the better-known tough climbs (classfied HC, or beyond categorisation) with elevation gains and average gradients.

Alpe d’Huez – 1150 metres @ 7.6% average
Col du Tourmalet
– 1395 metres @ 7.5% average (west side), 1250 metres @ 7.3% (east side)
Mont Ventoux
– 1620 metres @ 7.1% average
Col du Galibier
– 1580 metres @ 6.2% average
– 1580 metres @ 6.2% average
Col de la Croix-de-Fer
– 1520 metres @ 5.1% average
Col d’ Izoard
– 1200m @ 5.7% average (south side), 1500m @ 5% average (north side)
Hautacam
– 1170m @ 6.8% average
Col d’ Aubisque
– 1225 metres @ 4.2% average

Here are some of the toughest Tour de France stages in recent years:
Stage 17, 2010
– finish up the Col du Tourmalet
Stage 17, 2009
– Col de la Colombière via Col de Romme
Stage 17, 2008
– up Col du Galibier, Col de la Croix de Fer, with the finish up Alpe d’Huez.

## July 22, 2010

### A beautiful mathematical equality

Filed under: mathematics — ckrao @ 10:31 am

Here is an equality worth sharing, found at the CTK Insights blog here.

$\sqrt[3]{2 + \sqrt{5}} = \frac{1+\sqrt{5}}{2}$

I love the observation by Osler in [1] that the above number is constructible while the seemingly simpler $\sqrt[3]{2}$ is not.

Reference:
[1] T. J. Osler, Cardan Polynomials and the Reduction of Radicals, Math Magazine, Vol. 74, No. 1 (Feb. 2001), pp. 26-32

## July 18, 2010

### Fewest behinds in an AFL game

Filed under: sport — ckrao @ 3:40 am
Tags:

In the Australian Football League, July 17 2010 saw the Brisbane Lions score just one behind (Hawthorn 17.16 118 d Brisbane 7.1 43). This prompted me to find the VFL/AFL games with a team scoring the lowest number of behinds. Using the AFL tables here I came up with the following list. I found that such an incident has not happened since 1966! As expected most of the instances were in the early years of the competition.

0 behinds (5 times)
St Kilda 3.0 18 vs Geelong 14.12 96, Round 14 1897
St Kilda 1.0 6 vs Collingwood 8.17 65, Round 13 1898
St Kilda 1.0 6 vs Essendon 11.19 85, Round 9 1899
Hawthorn 7.0 42 vs Essendon 16.16 112, Round 12 1949
Fitzroy 1.0 6 vs Footscray 10.6 66, Round 5 1953

1 behind (24 times)
Geelong 3.1 19 vs Melbourne 9.10 64, Round 2 1897
Carlton 2.1 13 vs Essendon 5.12 42, Round 10 1897
Essendon 8.1 49 vs South Melbourne 6.9 45, Round 13 1897 (win)
St Kilda 3.1 19 vs Fitzroy 9.13 67, Round 10 1898
Carlton 1.1 7 vs Fitzroy 12.18 90, Round 16 1898 (Sectional Round 2)
Carlton 3.1 19 vs Melbourne 6.9 45, Round 3 1899
Geelong 3.1 19 vs South Melbourne 5.5 35, Round 6 1899
St Kilda 0.1 1 vs Geelong 23.24 162, Round 17 1899 (Sectional Round 3 – most lopsided game in history)
Essendon 2.1 13 vs South Melbourne 3.10 28, Round 17 1899 (Sectional Round 3)
Carlton 4.1 25 vs Collingwood 0.9 9, Round 5 1900 (win)
South Melbourne 3.1 19 vs Essendon 9.5 59, Round 15 1900 (Sectional Round 1)
St Kilda 3.1 19 vs Essendon 9.14 68, Round 2 1901
St Kilda 3.1 19 vs Collingwood 7.11 53, Round 7 1901
St Kilda 1.1 7 vs South Melbourne 7.14 56, Round 8 1901
Collingwood 5.1 31 vs South Melbourne 7.9 51, Round 10 1901
St Kilda 4.1 25 vs Carton 9.7 61, Round 11 1904
Fitzroy 3.1 19 vs Collingwood 5.11 41, Round 8 1905
Melbourne 2.1 13 vs Fitzroy 17.18 120, Round 15 1906 (Sectional Round 1)
St Kilda 5.1 31 vs Fitzroy 11.11 77, Round 16 1909
St Kilda 1.1 7 vs Carlton 5.15 45, Round 9 1915
North Melbourne 8.1 49 vs Melbourne 12.34 106, Round 7 1940*
Fitzroy 5.1 31 vs Geelong 20.24 144, Round 9 1966
Footscray 4.1 25 vs Carlton 12.14 86, Round 17 1966
Brisbane Lions 7.1 43 vs Hawthorn 17.16 118, Round 16 2010

* The Round 7 1940 game deserves a special mention since Melbourne scored a whopping 34 behinds in the same game! Only two games have seen more behinds scored by a team (Hawthorn with 41 vs St Kilda in Round 6 1977 and Geelong with 35 vs Melbourne in Round 16 1981).

No team has ever gone scoreless in a VFL/AFL game. The fewest behinds by a team in a finals game was 2 in the 1960 Grand Final, this 1909 Semi-final and this 2007 Qualifying Final. Two behinds have been scored by a team 67 times, most recently in Round 12 of the 2009 season.

Corrections are most welcome.

## July 9, 2010

### My Six Favourite Formulas

Filed under: mathematics — ckrao @ 10:38 am

At a later date I will explain them in further detail, but for now here they are!

• $\displaystyle e^{i\pi} + 1 = 0$
• $\displaystyle \int_M d\omega = \int_{\partial M}\omega$
• $\displaystyle \left[e^{{\mathbf a}.\nabla}\right]f({\mathbf x}) = f(\mathbf{ x+a})$
• $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$
• $\displaystyle \int_{-\infty}^{\infty} e^{-x^2}\ dx = \sqrt{\pi}$
• $\displaystyle \left(\sum_{n=1}^N n \right)^2 = \sum_{n=1}^N n^3$

## July 1, 2010

### Usain Bolt’s impact on the 100m world record

Filed under: sport — ckrao @ 10:45 am
Tags: ,

This graph says it all really.

I worked out by linearly extrapolating the world records prior to Bolt (back to 1983), that a time of 9.58 seconds should be reached by around 2027 – not 2009!

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