Brain Teasers
To keep your mind sharp
If you’ve been reading my blog/newsletter, you may have already sussed out a pattern I adhere to. That being posts released about food/nutrition, exercise, and mental acumen/health. In that sequence. Admittedly, I have not focused enough on the mental acumen part, rather, talking about general topics that can be an affront to one’s mental health, and thus overall wellness.
And wellness is the umbrella topic which all these subtopics I write about reside under.
Sharpening your intellect is often something which requires dedication and hard work. However, it need not be all the time. So in the following post, I shall propose a brain teaser which will get the ball rolling down this avenue. Stay with it, be patient, and push yourself to follow through until completion. At your own pace, but not a snail’s pace.
If you find yourself getting stuck, frustrated, or anxious do not worry. Just take a breather for a few hours, or overnight, and come back at it with a fresh mind.
Here we go.
Back in my Math and Physics days as a student I would often come across odd and even number generators. These held a small role in an otherwise much larger framework within what we were studying at the time. They would often be introduced in the following manner.
For n = 0,1,2,3,…
Even # Generator: 2n
Odd # Generator: 2n+1
Let’s unpack this a little bit. The variable (n) is listed as zero in addition to the set of positive integers. The ellipses (…) thingy indicates that the sequence of positive integers is to continue according to the observed pattern. And just as an fyi, the equals sign (=) in mathematics translates to the word ‘is’. So, For n = 0,1,2,3,… translates as follows.
For when n is chosen from the following sequence 0,1,2,3,…
Applying this to our even number generator, we will generate the set of even integers…of course.
n = 0 | 2(0) = 0 → That is 2 multiplied by 0.
n = 1 | 2(1) = 2 → That is 2 multiplied by 1.
n = 2 | 2(2) = 4 → That is 2 multiplied by 2.
n = 3 | 2(3) = 6
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And for the odd number generator:
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n = 0 | 2(0) + 1 = 1 → That is 2 multiplied by 0, plus 1.
n = 1 | 2(1) + 1 = 3 → That is 2 multiplied by 1, plus 1.
n = 2 | 2(2) + 1 = 5 → That is 2 multiplied by 2, plus 2.
n = 3 | 2(3) + 1 = 7
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OK. Now that we have that established, what I would like to do is introduce to you several different sequences of numbers. With each, see if you can pick up the pattern and find the next few numbers of the sequence. Then once you establish the pattern, try and synthesize it to a sequence generator for n = 0,1,2,3,…
In other words, given the sequence 0,3,6,9,12 you would eventually sort the next following numbers to be 15,18,21,24 and so on…ultimately arriving at the sequence generator of 3n.
Here they are:
2,3,4,5,…
5,6,7,8,…
3,5,7,9,…
1,4,7,10,…
2,7,12,17,…
0,2,6,12,…
0,6,16,30,…
0,3,8,15,…
The solutions will be added in the next brain teaser post. I hope to make this a regular series.
Now, for the best part about this I would like to address the question that math educators across the globe hear every day. ‘When will I even need to use this stuff…?’ Or, ‘When is this stuff ever used…?’
One huge use of this would be numerical data encryption. Since the dawn of the internet age, keeping communications safe and private between two point sources is of utmost importance. In fact, the internet itself (computer communication) dates back to the 1950’s I believe when the US Military essentially invented it. Data encryption was one tool used to keep messages safe and secure. Another fact, data encryption, or encryption generators dates back to WW2, where the US (and perhaps Germany) developed their own data encryption machines to keep type written paper letters encrypted should they get in the wrong hands.
The examples I’ve given here are just a small glimpse into this world and intended as a mental exercise to help keep you brains sharp over time. But, there is FOR SURE a major purpose, or use to this. Data encryption is utilized by all of our computers and smartphones constantly.
