What Equals 15 – The magic constant or magic sum of a magic square is the sum of the numbers in any row, column or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a regular magic square of order n, that is, a magic square containing the numbers 1, 2, …, n
For regular magic squares of orders n = 3, 4, 5, 6, 7 and 8, the magic constants are, respectively: 15, 34, 65, 111, 175 and 260 (order A006003 in OEIS). For example, a regular 8 × 8 square always equals 260 for each row, column, or diagonal. A simple magic constant of order n is n3 + n / 2. The largest magic constant of a simple magic square is also:
What Equals 15
Note that 0 and 1 are the only common magic constants in rational order, which are also rational squares.
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However, there are also infinitely many rational triangular numbers, rational generalized pentagonal numbers, and rational quadratic numbers, which are magic constants of rational order.
The term magic constant or magic sum applies similarly to other “magic” figures such as magic stars and magic cubes. The numerical shapes of the triangular grid are divided into equal polyworld areas containing equal sums to give the magic polyworld constant.
In 2013 Dirk Kinnes discovered the magic polytope series. The number of unique sequences that form the magic constant is now known up to n = 1000.
This model has two important features. It first demonstrates the balanced nature of all magic squares. If such a model is suspended from a central cell, the structure is balanced. (Consider the magic row/column sums .. equal mass equal distance balance). The second property that can be calculated is the moment of inertia. Adding the individual moments of inertia (cter × distance to the square of the cell value) gives the magic square the moment of inertia, which depends only on the order of the square.. These two basic concepts lead to more complexity the deeper you go into the subject. I find that using math, specifically algebra, makes it easier for me to figure out what I see when I work with Scala and Elm.
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Let’s start with the concept of immutability. If something is immutable, it means that it does not change. On its own, this doesn’t sound like much, but when applied to a programming paradigm it becomes very powerful. Let’s consider an example:
You change it. You can’t do that with languages like Elm, which is a functional language.
When I first encountered immutability, I found it hard to justify why it was so important. This is where algebra comes in.
(At least not in a way that doesn’t require rethinking how “numerically” is defined. But that should be left to math textbooks.)
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In your ordinary, non-functional program, not using immutable values is equivalent to always changing the value.
In this algebra problem. This increases the readability of errors and clouds. A good way to write a Scala example is as follows:
, can now be used if you want to make a more complex program. For example, you can say hello.
If someone approaches the greeter, the greeter can say “Hello” and if they leave the greeter, they can say “Goodbye.” So if someone runs past the greeter, they can expect the greeter to say both.
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If the constant is the potatoes of this operation, the purest functions are the meat (or tofu for plant eaters). As before, I think it’s helpful to go back to algebra to understand what pure functions are. An example like this really helped me visualize them:
By defining a pure function that returns a predictable value every time, because the same input should always return the same output, suddenly there’s not much left to know.
Debugging becomes easier and testing extreme cases is no longer a herculean task. You’ll feel better too
You can conclude that pure functions, their composition is also pure. It will give you more security when solving this problem.
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The same goes for function composition in FP! If you’re working with two pure functions, when you compose them, you can guarantee the same output given the same inputs.
As mentioned above, good FP is algebraic. So when learning why the fundamentals of FP are so important, it’s helpful to look at the math behind them.
I hope it rekindles the joy you had in your math classes over the past few years. If not, know that you can enjoy programming without looking back at those turbulent times. 😆
Software design and development requires the ability to collaborate, the ability to solve problems quickly. When a team is distant, productivity is reduced. Guide teams to collaborate in a remote culture.
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