The Game of Mu Torere

The New Zealand game of mū tōrere is illustrated above with a beautiful handmade wooden board. The game seems to have been developed by the Māori people in response to the European game of draughts (checkers). Play is quite different from draughts, however. The game starts as shown above, with Black to move first. Legal moves involve moving a piece to an adjacent empty space:

  • along the periphery (kewai), or
  • from the centre (pūtahi) to the periphery, or
  • from the periphery to the centre, provided the moved piece is adjacent to an opponent’s piece.

Game play continues forever until a draw is called (by mutual consent) or a player loses by being unable to move. Neither player can force a win, in general, so a loss is always the result of a mistake. For each player there is one “big trap” and two “small traps.” This is the “big trap” (Black wins in 5 moves):

The board on the left is the “big trap” for White – Black can force a win by moving as shown, which leaves only one move for White.

Again, Black moves as shown, which leaves only one move for White.

Now, when Black moves as shown, White cannot move, which means that White loses.

Here is one of the two “small traps” for White. The obvious move by Black results in White losing (but avoiding this does not require looking quite so far ahead as with the “big trap”):

Here (click to zoom) is the complete network of 86 game states for mū tōrere (40 board positions which can occur in both a “Black to move” and a “White to move” form, plus 6 other “lost” board positions). Light-coloured circles indicate White to move, and dark-coloured circles Black to move, with the start position in blue at the top right. Red and pink circles are a guaranteed win for Black, while green circles are a guaranteed win for White. Arrows indicate moves, with coloured arrows being forced moves. The diagram (produced in R) does not fully indicate the symmetry of the network. Many of the cycles are clearly visible, however:


Numeral Systems

Following up on my “origins of the alphabet” chart, here is one for numerals. The chart was produced using R, and the pictures are purely illustrative – unlike the pictures in the alphabet chart, they do not relate to the origins of the symbols.

Religion in the Australian Census

Following up on my earlier post, here is a chart of religion in Australia, by age (as per the 2016 Census, with percentages on the vertical axis relating to the population of Australia as a whole, and excluding people with no religion specified). Coloured areas in this chart indicate the total number of people for each religious group:

The changing religious landscape is revealed by the variation with age. For people aged 65, the population is 25% Catholic, 24% secular, 22% Anglican, 16% other Christian, 7% Uniting Church, 2% Buddhist, 1% other religion, 1% Muslim, and 1% Hindu.

For people aged 25, it’s 47% secular, 21% Catholic, 11% other Christian, 8% Anglican, 4% Muslim, 3% Hindu, 3% Buddhist, 2% Uniting Church, and 2% other religion. The chart below shows these relative percentages, for each age cohort.

Immigration and children are keeping the Catholic Church stable in size, but the Uniting Church is in collapse, and the Anglican Church is not doing much better (other data suggests that it’s in collapse outside of Sydney). The “big three” non-Christian religions (Islam, Hinduism, and Buddhism) are more than 10% of the age-25 demographic. The chart also shows the impact of student-driven Indian immigration to Australia over the past decade or so – there is a visible peak for Hinduism around age 33.

There seems to be something odd about the religion given for young children up to age 13 or so – some parents (especially Catholics) seem to be listing young children as “no religion.” This might reflect delayed baptism. However, it also seems that many children lose their childhood religion in late teens and early adulthood.

Mean ages for adults within the different groups are Hindu: 37.1, Muslim: 37.7, secular: 42, other religion: 42.6, Buddhist: 43.5, Catholic: 48.6, other Christian: 50.4, Anglican: 54.8, and Uniting Church: 55.8. The last two groups in particular are skewed towards older people.

Colours in national flags

The infographic above shows the most common colour in various national flags, excluding white and black. For example, red is the most common colour in the US flag. If there are two or more equally common colours (as in BE = Belgium or FR = France), the country is given partial credit for both. Similar colours are grouped using k-means clustering in R.

Overall, shades of red seem the most popular, followed by shades of blue. The set of flag image files I analysed wasn’t fantastic, however, and that may have skewed the results slightly.

Democracy, Religion, and Same-Sex Marriage in Australia

The results of the postal survey are in, and Australia has voted 61.6% “Yes” to same-sex marriage. Or rather, it seems that two Australias voted. The official results have been made available by electorate, which means that they can be correlated with demographic factors (and my readers know that I love doing that). The average age of each electorate had no effect, but religious composition certainly did.

According to the 2016 census, Australia’s stated religious composition looks like this (where the 33.3% “Secular” includes Agnostic, Atheist, Humanist, New Age, and Unitarian Universalist):

The chart below shows a strong correlation (0.82) between the percentage of “Secular” people in an electorate, and the size of the “Yes” vote. If all the “Secular” people voted “Yes” (as seems likely), this means that 58% of the religious people voted “No.” Doing some simple multiple linear regression, there was a statistically significant link between religion and voting “No” for every major religious group. This link was strongest for Muslims, Hindus, Buddhists, Orthodox, the Uniting Church, and other non-Anglican Protestants. It was a little weaker for Anglicans and even more for Catholics, although the Anglican link was quite strong in Victoria, South Australia, New South Wales, and Queensland. The Catholic link was quite strong in the last three of those states.

Electorates in the chart are coloured according to the largest religious group within them. Sydney is 52.7% Secular, for example (as well as 8.6% Buddhist, 1.7% Muslim, 1.7% Hindu, 1% Jewish, 17.9% Catholic, 2.4% Orthodox, 13.5% Protestant, and 0.5% Other Religion). It voted 83.7% “Yes.”

Blaxland is 32.2% Muslim (as well as 9% Buddhist, 3.3% Hindu, 21.2% Catholic, 5.5% Orthodox, 13.2% Protestant, 0.7% Other Religion, and 14.9% Secular). It voted 73.9% “No.”

McMahon is 39% Catholic (as well as 5.9% Buddhist, 12.4% Muslim, 2.9% Hindu, 6.9% Orthodox, 18.5% Protestant, 1.4% Other Religion, and 13.2% Secular). It voted 64.9% “No.”

Barton is multi-religious with 28.1% Secular being the largest group (as well as 5.6% Buddhist, 8.4% Islam, 5.6% Hindu, 0.2% Jewish, 22.6% Catholic, 15.7% Orthodox, 13.3% Protestant, and 0.5% Other Religion). It voted 56.4% “No.”

It does seem that there is a secular Australia, which voted overwhelmingly “Yes,” and a religious Australia of twice the size, which voted mostly “No.” If the disparate religious communities in Australia realise that they have more in common than they have thought, that could have quite a significant influence on Australian politics in the future.

A (distorted) geographical view of the postal survey results

Guns, education, religion, and suicide

My earlier post indicated that gun laws in the US had little impact on the homicide rate, when demographic factors were taken into account. This makes sense – if I want to kill somebody, the lack of a gun will merely prompt me to choose another weapon. But what about suicide? The impulse to suicide is often brief, and easy access to a gun during a suicidal episode may increase the chance of dying.

To test this, I extended my previous dataset with data on educational attainment, data on religiosity, registered gun ownership data from the ATF, age-adjusted suicide rates from the CDC, poverty rates, unemployment rates, and other demographic data. I ran all that through a regression tree analysis, using R.

Suicide rates in the chart (click to zoom) are indicated by colour, ranging from 8 per 100,000 for New Jersey and New York (yellow) to 23.7 for Montana (black). Having a college degree seems to have a protective effect – states on the right of the chart, with more college degrees, had lower suicide rates. This may relate to the higher employability of college graduates. However, states at the top of the chart, with higher high school graduation rates, had higher suicide rates. I am not sure why this is the case.

Among the states with fewer college graduates, religion had a protective effect (this is consistent with other studies). States where 77% or more of the population said that religion was “somewhat important” or “very important” to them are indicated on the chart by triangles. For the states with fewer college graduates, the suicide rate was 13.6 per 100,000 for religious states, and 17.5 for less religious ones.

Finally, the highest-risk states (fewer college graduates and less religious) split according to gun ownership. States with more than 0.008 registered guns per capita are marked on the chart with an inner dot. Among the highest-risk states, the suicide rate increased from 13.9 per 100,000 to 18.6 when more guns were present. This group included Alaska (23.2 per 100,000), Arizona (17.5), Idaho (19.2), Maine (17), Montana (23.7), Nevada (18.6), North Dakota (17.3), Oregon (16.8), and Utah (21.4). Among the more religious states, registered gun ownership did not seem to have an effect (although, of course, registered gun ownership is a poor indicator of true gun ownership).

Thus the data does seem to suggest a link between gun ownership and suicide risk, but only when other risk factors are present (low religiosity and no college degree). This is exactly what we expected, and it means that suicidal (or potentially suicidal) people need to be kept away from guns.

Do gun laws save lives?

Somebody pointed me at this interesting data the other day. The chart above (click to zoom) combines the “Gun Law Score Card” from the Law Center to Prevent Gun Violence in the US with homicide rate data from Wikipedia and voting results from the last US election. Do gun laws reduce the chance of being murdered?

Obviously, “Blue” states tend to have stricter gun laws than “Red” states (an average of B− vs D−). “Blue” states also have lower homicide rates than “Red” states (4.5 vs 5.9), and this is statistically significant (p = 0.012). There is a weak (R2 = 6%) correlation between gun laws and homicide rates, but this relationship is not statistically significant.

Whatever it is that makes you less likely to be murdered in some states than others, it does not primarily seem to be the gun laws. Poverty may be one of the relevant factors, however – median household income explains 22% of the variance in homicide rates, and when this is taken into account, any effects due to gun laws or election results disappear. “Red” states are, on the whole, simply poorer (and, conversely, poor states are more likely to vote Republican and have weak gun laws). Other demographic factors, such as the number of people with college degrees, also seem to have explanatory value as far as the murder rate is concerned. However, the phenomenon of murder does not seem to be understood as well as it could be.