# Flow

Judson Brewer, an addiction psychiatrist and neuroscientist at Yale University, describes flow and his research findings on the topic.

Video sourced from YouTube here.

# The New World of Sales

Author Daniel H. Pink answers seven questions about pitching, selling, and closing for the 21st Century workforce.

Video sourced from YouTube here.

# Blohg to Pelican

Over the past few days i moved this website's backend from Blohg to Pelican, both of which are RST-capable website generators. I did so, because i don't need the dynamic Flask-application capabilities of Blohg and because Pelican is more mature and equally simple to use.

The transition was mostly smooth, but i did encounter a few issues that required me to search beyond Pelican's documentation to resolve. To save new Pelicaneers the effort, here are those issues and resolutions.

**Issue 1**: Custom paginated templates.

I wanted a separate landing page and blog page, and the Pelican docs don't elaborate on this point.
Querying the Pelican developers through the project's Github page, i received instructions from justinmayer.
First, he said, i needed to make a custom template to house my blog, which i called `blog.html`.
It contains the code

```
{% extends "base.html" %}
{% set active_page = "blog" %}
{% block title %}{{ SITENAME }} - Blog{% endblock %}
{% block content %}
{% for article in (articles_page.object_list if articles_page else articles) %}
<div class="blogItem">
<h1><a href="{{ SITEURL }}/{{ article.url }}">{{ article.title }}</a></h1>
{{ article.content }}
<div class="blogMeta">
Author: <a href="mailto: {{ AUTHOR_EMAIL }}">{{ article.author }}</a><br>
Date: {{ article.locale_date }}<br>
{% if article.tags %}
Tags:
{% for tag in article.tags %}
<a href="{{ SITEURL }}/{{ tag.url }}">
{{ tag }}</a>{% if not loop.last %}, {% endif %}
{% endfor %}<br />
{% endif %}
<a href="{{ SITEURL }}/{{ article.url }}#disqus_thread">Comments</a> -
<a href="{{ SITEURL }}/{{ article.url }}">Permalink</a>
</div>
</div><!-- end #blogItem -->
{% endfor %}
{% include 'pagination.html' %}
{% endblock content %}
```

By the way, here's the code for the pagination template `pagination.html`:

```
{% if articles_page and articles_paginator.num_pages > 1 %}
<div class="pagination">
<ul>
{% if articles_page.has_previous() %}
{% set num = articles_page.previous_page_number() %}
<li class="prev"><a href="
{{ SITEURL }}/{{ page_name }}{{ num if num > 1 else ''}}.html"
>«</a></li>
{% endif %}
{% for num in range( 1, 1 + articles_paginator.num_pages ) %}
<li><a href="
{{ SITEURL }}/{{ page_name }}{{ num if num > 1 else '' }}.html"
class="{{ 'active' if num == articles_page.number else '' }}"
>{{ num }}</a></li>
{% endfor %}
{% if articles_page.has_next() %}
<li class="next"><a href="
{{ SITEURL }}/{{ page_name }}{{
articles_page.next_page_number() }}.html"
>»</a></li>
{% endif %}
</ul>
</div>
{% endif %}
```

I got it from the tuxlit_tbs Pelican theme and modified it to my liking.

Second, i had to add the following lines to my `pelicanconf.py`.

```
DIRECT_TEMPLATES = (('index', 'blog', 'tags', 'categories', 'archives'))
PAGINATED_DIRECT_TEMPLATES = (('blog',))
```

**Issue 2**: Disqus comments.

This is another topic the Pelican docs don't elaborate on. To embed Disqus comments in my blog pages, i read Disqus's help article on embedding comments and Disqus's help article on adding comment counts and did the following.

First, i made the template `disqus_comments.html` which contains the code

```
{% if DISQUS_SITENAME %}
<div class="blogItem">
<h2>Comments</h2>
<div id="disqus_thread"></div>
<script type="text/javascript">
var disqus_shortname = '{{ DISQUS_SITENAME }}';
(function() {
var dsq = document.createElement('script');
dsq.type = 'text/javascript';
dsq.async = true;
dsq.src = 'http://' + disqus_shortname + '.disqus.com/embed.js';
(document.getElementsByTagName('head')[0] ||
document.getElementsByTagName('body')[0]).appendChild(dsq);
})();
</script>
<noscript>
Please enable JavaScript to view the
<a href="http://disqus.com/?ref_noscript={{ DISQUS_SITENAME }}">
comments powered by Disqus.
</a>
</noscript>
<a href="http://disqus.com" class="dsq-brlink">
blog comments powered by <span class="logo-disqus">Disqus</span>
</a>
</div>
{% endif %}
```

and in my `article.html` template added the line `{% include 'disqus_comments.html' %}` before the line `{% endblock %}`.
Doing so embeds Disqus comments for each blog post (article).

Second, i made the template `disqus_comment_counts.html` which contains the code

```
{% if DISQUS_SITENAME %}
<script type="text/javascript">
var disqus_shortname = '{{ DISQUS_SITENAME }}';
(function () {
var s = document.createElement('script'); s.async = true;
s.type = 'text/javascript';
s.src = 'http://' + disqus_shortname + '.disqus.com/count.js';
(document.getElementsByTagName('HEAD')[0] ||
document.getElementsByTagName('BODY')[0]).appendChild(s);
}());
</script>
{% endif %}
```

and in my `base.html` template added the line `{% include 'disqus_comment_counts.html' %}` before the `</body>` tag.
That enables blog post comment counts via the tags `<a href="{{ SITEURL }}/{{ article.url }}#disqus_thread">Comments</a>`.

**Issue 3**: Typesetting math.

I wanted to use MathJax to typeset math on this site.
Pelican has a LaTeX plugin, which uses MathJax, but doesn't work properly at present.
It didn't typeset math in the list view of my blog posts, only in the detail view of each post.
So i searched the web for an alternative and found this helpful blog post from another Pelican user.
Following its instructions, i added this Javascript code into the `<head>` section of my `base.html` template:

```
<!-- Using MathJax, with the delimiters $ -->
<!-- Conflict with pygments for the .mo and .mi -->
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
"HTML-CSS": {
styles: {
".MathJax .mo, .MathJax .mi": {color: "black ! important"}}
},
tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']],processEscapes: true}
});
MathJax.Hub.Register.StartupHook("HTML-CSS Jax Ready",function () {
var VARIANT = MathJax.OutputJax["HTML-CSS"].FONTDATA.VARIANT;
VARIANT["normal"].fonts.unshift("MathJax_SansSerif");
VARIANT["bold"].fonts.unshift("MathJax_SansSerif-bold");
VARIANT["italic"].fonts.unshift("MathJax_SansSerif-italic");
VARIANT["-tex-mathit"].fonts.unshift("MathJax_SansSerif-italic");
});
MathJax.Hub.Register.StartupHook("SVG Jax Ready",function () {
var VARIANT = MathJax.OutputJax.SVG.FONTDATA.VARIANT;
VARIANT["normal"].fonts.unshift("MathJax_SansSerif");
VARIANT["bold"].fonts.unshift("MathJax_SansSerif-bold");
VARIANT["italic"].fonts.unshift("MathJax_SansSerif-italic");
VARIANT["-tex-mathit"].fonts.unshift("MathJax_SansSerif-italic");
});
</script>
<script type="text/javascript"
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML">
</script>
```

I removed the `['$','$']` item which allows dollar signs to delimit math text, as is standard in LaTeX documents, but it requires escaping every backslash in the math text with an additional backslash.
For now i'll stick with the standard RST role `:math:` for inline math and the standard RST directive `.. math::` for block math display.
The area of a circle is *A*_{c} = (*π* ⁄ 4)*d*^{2}, that is,

*A*

_{c}= (

*π*)/(4)

*d*

^{2}.

# Delicious Dal

## Ingredients

- 1.5 C green or brown lentils
- 5 C water
- 2 T coconut oil
- 1 400mL can chopped tomatoes
- 1 onion chopped
- 2 cloves garlic chopped
- 1 T shredded ginger
- 1 t ground turmeric
- 1 hot chili
- 1 red capsicum chopped
- 1 t salt
- 1 t cumin seeds
- 2 t black mustard seeds
- some coriander leaf for garnish

## Directions

- Except for 1 T coconut oil, the cumin seeds, and the mustard seeds, put everything into a pressure cooker, and cook at high pressure for 11 minutes.
- Fry the reserved coconut oil, cumin seeds, and mustard seeds until the mustard seeds begin to pop.
- Add the fried seeds and oil to the soup, garnish with coriander leaf, and serve.

## Notes

Serves about 6. C = cup, T = tablespoon, t = teaspoon.

# Store Credit

Here's the store credit problem from Google Code Jam Qualification Round Africa 2010, which i discovered via Programming Praxis:

You receive a credit C at a local store and would like to buy two items. You first walk through the store and create a list L of all available items. From this list you would like to buy two items that add up to the entire value of the credit. The solution you provide will consist of the two integers indicating the positions of the items in your list (smaller number first). For instance, with C=100 and L={5,75,25} the solution is 2,3; with C=200 and L={150,24,79,50,88,345,3} the solution is 1,4; and with C=8 and L={2,1,9,4,4,56,90,3} the solution is 4,5.

## Algorithm 1

Brute force. Test all pairs in the list. O(n^2) time and O(n) space.

## Algorithm 2

Sort the list and then iterate over the list, at each step looking for the complementary price of the current price via binary search. O(n lg n) time and O(n) space.

## Algorithm 3

Store the prices and indices in a dictionary and then iterate over the dictionary, at each step looking for the complementary price. Since dictionary look-up is constant time on average, this is an O(n) time (in the average case) and O(n) space algorithm. Hooray!

Here's a Python 2.7 implementation:

```
def find_pair(credit, prices):
# Make a dictionary with items of the form
# (price, set of indices of items with that price).
# Note that the price list could contain duplicate prices.
d = dict()
for (i, p) in enumerate(prices):
d[p] = d.get(p, set())
d[p].add(i + 1) # + 1 to index from 1 instead of 0.
for price in d.keys():
if price <= credit:
# Remove price index from d.
i = d[price].pop()
# Check whether there's a complementary price in d.
if credit - price in d:
# This works even if price == credit - price.
j = d[credit - price].pop()
return min(i, j), max(i, j)
# Reinsert price index into d.
d[price].add(i)
return "Sorry, no pair of item costs sum to %s" % credit
```

# Cat Vs. Dog

Updated for clarity 23 Sep 2013.

A friend of mine recently pointed me to Spotify's Cat vs. Dog puzzle. The question is long, and so i'll summarize it without its motivating context.

You are given n votes, each of which is a pair of strings of the form (*Ci*, *Dj*) (a cat-lover vote) or (*Dj*, *Ci*) (a dog-lover) for some integers 1 ≤ *i*, *j* ≤ *n*.
This collection of votes can contain repeats.
Two votes (*a*, *b*) and (*c*, *d*) are said to conflict if *a* = *d* or *b* = *c*.
So a conflict can only occur between a cat-lover vote and a dog-lover vote.
Your job is to find the size of a maximal subcollection of the votes in which no two votes conflict.

## Algorithm 1

Brute-force. Run through all possible subcollections of votes and test each for conflicts. Choose the biggest non-conflicting subcollection.

This procedure would take *O*(2^{n}) time in the worst case, because there are 2^{n} subcollections to check.
That's very slow.
Can we be more clever?

## Algorithm 2

Yes, model the problem with a graph.

- Distinguish the votes by tagging each with a unique identifier, such as an integer between 1 and
*n*. This takes*O*(*n*) time. - Build the following undirected graph
*G*. Make each vote a vertex, and add the edge {*u*,*v*} if*u*and*v*are conflicting votes. This takes*O*(*n*^{2}) time. Notice that*G*is bipartite, because every edge connects a cat-lover vote to a dog-lover vote and no vertex is both a cat-lover vote and a dog-lover vote. - Notice that every maximal subset of nonconflicting votes is the complement of a minimum vertex cover for
*G*. So find the size*k*of a minimum vertex cover for*G*and return*n*−*k*. Now, finding the size of a minimum vertex cover in an arbitrary undirected graph is an NP-complete problem, and there's no known sub-exponential time way to do that. However, our graph is bipartite, and so by Koenig's theorem, the size of a minimum vertex cover of*G*equals the size of a maximum matching for*G*, the latter of which can be found by the Hopcroft-Karp algorithm in*O*(|*E*|√(*n*)) time, where |*E*| is the number of edges in*G*. In the worst case, |*E*| =*n*^{2}and running Hopcroft-Karp takes*O*(*n*^{2.5}) time.

So all up, Algorithm 2 takes *O*(*n*^{2.5}) time in the worst case.

# Contours: A Mathematical Coloring Book

Here is a mathematical coloring book I recently made. It contains black and white pictures generated by mathematical formulas which you color in for fun. Best printed on size A4 paper or bigger. Enjoy and share!

# Jumping Jack

I'd like to present some solutions to the following puzzle that appeared as March 22, 2013 Programming Praxis puzzle.

Jack starts at 0 on the number line and jumps one unit in either direction. Each successive jump he makes is longer than the previous by 1 unit, and can be made in either direction. Design an algorithm that takes an integer n and returns a shortest sequence of jumps that Jack can perform to reach n along with the length of that sequence.

## Algorithm 1

Brute-force. Perform a breadth-first search through the binary tree of all valid jump/integer summand sequences (0, -1, 1, -1 - 2, -1 + 2, 1 - 2, 1 + 2, ...) and stop when the target sum n is reached.

## Algorithm 2

Brute-force with pruning. Label the nodes of the binary tree with their partial sums and order the nodes of so that the left child is less than the right child (0, -1, 1, -3, -1, -1, 3, ...). Notice that every pair of nodes at the same height with equal partial sums have equal subtrees. Notice also that every pair of nodes at the same height with partial sums of opposite sign have mirror image subtrees. Notice also that flipping the signs/turns of a path with partial sum s yields a path of the same length to -s. So in our breadth-first search, we don't need to search down a node at a given height if we've already discovered a node at the same height with the same magnitude partial sum.

Analyzing the worst-case time complexity of Algorithm 1 or Algorithm 2 requires knowing at what tree height the given number n is first discovered. If you can figure that out, then you probably discovered...

## Algorithm 3

Smart-force. Find the least integer k such that t_k := 1 + 2 + 3 + ... + k >= |n| and t_k has the same parity (evenness/oddness) as |n|. Then flip the signs of the summands of t_k in a systematic way to total n.

Here are the details. Notice that t_k = k(k + 1)/2, a triangular number. Setting t_k = |n| and solving for k gives the ceiling of (-1 + sqrt(1 + 8 |n|))/2. So k equals the ceiling or 1 or 2 plus it, whichever of those three numbers has the same parity as n and is least. Here we're using the fact that the set {t, t + s, t + s + (s + 1)} of three consecutive triangular numbers contains both even and odd numbers for any positive integers t, s. (Simply check all four parity possibilities for t, s.)

To find a minimal length sum for n, first compute d := (t_k - n)/2. Because t_k >= |n| and t_k and n have the same parity, d lies in the set {0, 1, 2, ..., t_k}. Now repeatedly subtract: d = a_k (k) + r_k, r_k = a_{k-1} (k-1) + r_{k-1}, ..., r_2 = a_1 (1) + r_1, choosing each a_i maximally in {0, 1}. By the lemma below, r_1 = 0. So d = sum_{i=1}^k a_i i. Thus n = t_k - 2d = sum_{i=1}^k i - sum_{i=1}^k 2a_i i = sum_{i=1}^k (1 - 2a_i) i and 1 - 2a_i in {-1, 1}. So the sequence b_i := 1 - 2a_i is a path, and by the minimality of k, b_i is a minimal path.

## Algorithm 3 example

Consider the target number n=12. According to Algorithm 3, the possibilities for k are 5, 6, 7. The corresponding values of t_k are 15, 21, and 28. Since 28 is the least of these with the same parity as n, we see that k=7. So d = (t_k - n)/2 = 8, which we write as 1 + 7 according to the algorithm. Thus a shortest path to 12 is -1 + 2 + 3 + 4 + 5 + 6 - 7.

I say *a* shortest path, because shortest paths aren't unique in general.
For example, 1 + 2 -3 + 4 - 5 + 6 + 7 also works.

## Algorithm 3 correctness

*Lemma*: Let A_k = {0, 1, 2, ..., t_k}.
Then a number lies in A_k if and only if it can be expressed as a sum sum_{i=1}^k a_i i for some a_i in {0, 1}.

*Proof*: By induction on k.
First, 0 = sum_{i=1}^0 1, the empty sum.
Now suppose the result holds for all k - 1 >= 0 and let d in A_k.
Repeatedly subtract: d = a_k (k) + r_k, r_k = a_{k-1} (k-1) + r_{k-1}, ..., choosing each a_i maximally in {0, 1} and stopping at the first r_j in A_j for some j < k.
Then by the induction hypothesis, r_j = sum_{i=0}^j b_i i for some b_i in {0, 1}.
Then d = r_j + sum_{i=j+1}^k a_k i, as desired.
Conversely, a sum s := sum_{i=1}^k a_i i for a_i in {0, 1} satisfies 0 <= s <= sum_{i=1}^k i = t_k, and so s in A_k.

## Algorithm 3 time complexity

Assuming that arithmetic operations are constant time, it takes O(1) time to compute k from n, and hence the length of a minimal path to n. Then it takes O(k) time to find a minimal length sum for d and O(k) time to use that sum to produce a minimal length sum for n. So O(k) = O(sqrt(n)) time all up.

# Hourly Wage

Suppose you work in New Zealand, would like to earn a given net weekly income (for 52 weeks per year) after income tax, and would like to work a given number of hours per week (for 47 weeks per year; NZ workers get 4 weeks annual leave plus roughly 1 week of public holidays). What gross hourly income must you earn?

To answer that question, i used the 1 April 2012 to 31 March 2013 NZ income tax rates for individuals, wrote this Python program, and used it with Sage to make this table:

The gross hourly incomes needed are displayed in the main body of the table in New Zealand dollars, rounded to the nearest dollar and excluding GST. For example, to earn $1200/week after tax, working 25 hours/week, you need to earn $68/hour before tax.

# Science and Python: retrospective of a (mostly) successful decade

Video sourced from YouTube here.

**Why no comments?**
I used to do public comments but found that moderating and maintaining them took too much time in front of the computer, time better spent playing outdoors.
So these days I only do private comments, that is, you can email me comments regarding a post by clicking the 'Comment' link at the bottom of the post.