Using math puzzles to illustrate the nature of science

(As I mentioned some time ago, I am working on my next book that is tentatively titled The Paradox of Science. From time to time, I will try out ideas from it on the blog, suitably modified to make the blog posts self contained. Readers get the benefit of a sneak preview and I hope to get feedback from readers as to clarity, correctness, style, etc. Note that the book is aimed at the interested layperson and not the many experts who read this blog so put yourself in their shoes when reading. This post is the first of such offerings. Enjoy!)

The website Fivethirtyeight has a weekly feature called The Riddler where puzzles of a mathematical sort are presented and the solution given the following week. Here is one such problem stated in its entirety:

Complete this series:

10, 11, 12, 13, 14, 15, 16, 17, 21, 23, 30, 33, …

I want to use this simple purely mathematical puzzle to illustrate an important insight about science. I will give the solution below the jump and then discuss the relation to science.
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Muhammad Ali (1942-2016)

Muhammad Ali died yesterday and the tributes are pouring in. His greatest accomplishment may have been that he persuaded so many people (including me) to become fans of a brutal activity (I cannot call it sport) that should be despised out of existence. But even more than his boxing skills, what electrified people like me around the world who detested what the US was doing to Vietnam was what he said when he refused to be drafted to fight in that horrible war.
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The rise of physical and verbal violence

I expressed concern yesterday over the violence between political partisans that seem to be becoming a routine feature of this election. This is not a healthy sign. It is interesting how politics in US are a delayed mirroring of what I was used to in Sri Lanka. There too violence at political rallies was not uncommon. Another ugly feature there were the attempts by politicians to impugn the integrity of judges and even retaliate against them or threaten them and thus undermine their credibility or intimidate them whenever they ruled against the government in power.
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Ethical issues with experimental drug treatments

The US has a pretty strong regimen for getting drugs approved, a system that has great credibility after it kept out the nausea and morning sickness combatting drug thalidomide that in the 1950s wreaked havoc on babies in other countries. The gold standard for new drugs to be allowed to be prescribed is, as I understand it, that they have to show that they do not pose significant risks and also undergo double-blind clinical trials on humans that demonstrate that they work better than a placebo. The process is long and expensive.
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The agony of conservative intellectuals

As Donald Trump tries to get the Republican party to unify around him, there has been some shifting around among the members of the various sectors within the right-wing party, such as the elected officials, the media, and the influence shapers. One is seeing the first two sectors falling into line behind Trump as they see it in their best interests to not defy the party nominee.
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Technology is not going to save religion

As we all know, young people are leaving religious institutions in ever-increasing numbers. This is naturally causing concern in religious quarters and some wealthy people think that the problem is that the young are not being equipped to ‘properly’ answer the kinds of deep questions that occur to almost everyone as they are growing up and becoming independent. By properly, these people mean consistent with what the Bible says.
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The latest example of how the US military can do no wrong

We have seen that no one in the US government ever faces serious punishments for the torture, death, and destruction of innocent people in the many countries with which we are at war, except for those who reveal the truth about such events, as was the case with the punishment meted out to Chelsea Manning for her release of the Collateral Murder video.
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