After recoiling and feeling vaguely nauseous at the flagrant misuse of either the addition sign, or equals sign, or both, you are conned into spending a little time working out what on earth is going on.
SPOILER ALERT: If you wish to work it out for yourself, do so before reading further.
Otherwise, scroll past the following mathemakitten.
You have probably realised that when you actually add the two numbers together, their total becomes the right-most digit(s) of the answer. Trying out the other mathematical operations will lead you to discover that the left-hand side of the answer is the result of subtracting the two numbers. So combining 6 and 4 gives 210 because 6-4=2 and 6+4=10.
It is far from the most devious inductive reasoning problem of this type and the solution can be understood by anyone with a grasp of basic arithmetic. The operation does, however, have an interesting feature: It is rather messy to produce a formula that gives the final answer in terms of any inputs a and b.
It is very tempting just to concentrate on maths that is sleek and elegant so it is good to remind ourselves that the numerical universe occasionally appears to have been dreamt up by Jackson Pollock.
Before further investigating the function behind this puzzle, we need to change how it is notated. The plus sign just will not do; we don't want to confuse it with the traditional meaning of the symbol. Instead, we'll ring it and write
This type of device is known as a binary operator; we put two numbers in and get one out. To begin with, we'll assume that we can only put in two distinct positive integers and we have to have the larger first (in order that the subtraction results in a positive integer). The output will be a third positive integer.
There are a number of questions we might ask: Can we produce all integers as a result of the operation? If not, which can't we get? Can we get the same number out for two different pairs of inputs?
The last of these questions is easiest to answer because we can find an example do so with an example. Consider the output 1317. We can either reach it with two numbers that differ by 13 and total 17, or by two consecutive integers that add to give 317:
This tells us that the function cannot be inverted; we cannot tell, just by looking at the output, what the input is. It's admittedly a small fact but isn't it satisfying to have already a slightly greater insight into the mathematics than the person who set the original problem?
How about numbers we can't get out? Well let's suppose we had an output made up of the difference X and the total Y of inputs a and b. Then the following simultaneous equations would hold:
Solving these for a and b gives
It is very tempting just to concentrate on maths that is sleek and elegant so it is good to remind ourselves that the numerical universe occasionally appears to have been dreamt up by Jackson Pollock.
Before further investigating the function behind this puzzle, we need to change how it is notated. The plus sign just will not do; we don't want to confuse it with the traditional meaning of the symbol. Instead, we'll ring it and write
There are a number of questions we might ask: Can we produce all integers as a result of the operation? If not, which can't we get? Can we get the same number out for two different pairs of inputs?
The last of these questions is easiest to answer because we can find an example do so with an example. Consider the output 1317. We can either reach it with two numbers that differ by 13 and total 17, or by two consecutive integers that add to give 317:
How about numbers we can't get out? Well let's suppose we had an output made up of the difference X and the total Y of inputs a and b. Then the following simultaneous equations would hold:
Solving these for a and b gives
Given that we have (for now) stipulated that a and b must be positive integers, we see immediately that when X and Y are not both even or not both odd, there is no input that will produce the corresponding output. So 12, 14, 16, 18, 23, 25, etc. cannot be produced.
We have quickly learnt a couple of things using some elementary analysis. To investigate the function further, however, it would be really useful to turn the output into an algebraic formula. This is where things begin to get murky since the size of the output seems very sensitive to small changes in the input. Operating on 5 and 4, for example, gives 19 but if we change the input slightly to 6 and 5, the answer leaps to 111.
This results in a formula that appears far more complicated than we might imagine it should. Perhaps I've overlooked something and made things much too tricky but the simplest way I could think of to calculate an output is
That strange pair of bottom-heavy square brackets is called the floor function. It simply rounds the number inside to the next integer down (numbers that are already integers remain as they are). This, combined with the logarithm, simply counts one fewer than the number of digits in a+b and provides a way of ensuring that the difference and sum don't overlap in the final answer. Those with a particularly keen eye for these things might wish to ask themselves why I have had to add one to the floor function rather than just using the ceiling function.
This formula can be entered into MS Excel so we are able to quickly produce a table of values for varying a and b. Below I have highlighted those which appear in the original problem
The nice thing about the formula is that it provides a natural extension to any pair of positive numbers.
We can now produce an outcome if b is bigger than a:
It could be argued that an answer of -210, rather than -190, might be more in keeping with the spirit of the original puzzle but it would require another complication in the formula. Furthermore, the question setter failed to leave any contact details on the original social media post so I have no idea whom to ask about his or her intention in these cases.
We can also extract results for non-integers. You might like to check the following (or just take my word for it):
Graphs are often useful for visualising functions. Let's see what happens if we fix a=8 and vary b.
 |
| Value of function (vertical axis) as b varies (horizontal axis) for a=8 |
Will this proceed on the same trajectory for ever?
No. The next dramatic thing will happen when b=92 (and then 992, 9992, 99992, etc.)
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| Value of function for a=8 showing step when b=92. |
We could play the same game in the other direction by fixing b and varying a.
It might be more fun, however, to demonstrate what happens as both vary. We can do this by plotting a surface whose height above two perpendicular horizontal axes will represent the function's output.
As you can see, for small values of a and b, the function is planar with a slight tilt. This takes a dramatic twist along its line of intersection with the plane a+b=10. If I were to plot it for larger values we would doubtless see another twist at a+b=100 and then a+b=1000 and so on. In my opinion, this surface reveals the patterns behind the original puzzle with far greater clarity than a large table of results can.
Many would question the point of embarking upon an exploration once the puzzle has been solved, expressing doubt about whether the results are "useful in everyday life". That is of absolutely no consequence to a mathematician, however, who just feels the compelling need to venture beyond the trivial.
What's more, there is always an extension or generalisation to be made; the job is never done. Can the formula be modified to allow any pair of real numbers? How about complex numbers? Is the function associative? Can we restrict its domain to form a group of some sort?
Right now, however, it's time for some tea and cake.
from matheminutes http://ift.tt/1ORXkjl
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