When there’s a tradition as broad and deep (and fascinating) as math, there are bound to be aficionados who are nuts about the subject—in other words, geeks. They are sure to compulsively and meticulously nitpick minor details that ordinary practitioners would miss.
Such math fans failed to miss the alleged quirk in the sequence given here (shown below). Reportedly, millions of people have been debating that there is more than one solution—insisting that there are in fact two “secret” solutions for the gifted among us to figure out.
Could they be right about that? Let’s see.
First, let’s try to figure out what the solution is for ourselves before delving into what throngs of math geeks are jabbering over. Observe the sequence shown below and try to solve the final equation based on the pattern that you see:
Take a moment or two to figure it out before checking for the answer.
You may have clued in from the first line (9 = 72) that it will jive if restated like this: 9 × 8 = 72. Similarly, line two will jive if restated like this: 8 × 7 = 56. From this we may infer the pattern: a × (a – 1) = x. We can test this by plugging it into the rest of the sequence, and we see that it works.
If plugged into the last line, we get: 3 × (3 – 1) = 6. The final answer is 6.
What are the Two “Secret” Solutions?
There are two more potential answers according to some math enthusiasts. Take a moment, if you like, to determine what those are, and then scroll down to see the answers.
“Secret” Solution 1: Decreasing Subtraction
You may have spotted a second pattern of “decreasing subtraction.” That is, an amount subtracted from the right side of an equation—an amount that decreases by 2 each subsequent equation in this case—gives us the right side of the subsequent equation. In this sequence, 16 is subtracted from the right side of the first equation giving us the right side of the second equation. Next, 16 – 2 = 14, which when subtracted from the right side of the second equation gives us the answer for the third equation, and so on.
Then, if we apply this sequence to the final equation, the final answer is 12.
“Secret” Solution 2: Decreasing Multiplication
Believe it or not, there yet is another geeks-only solution called “decreasing multiplication.” That is, as we have observed in the first line, 9 × 8 = 72. In the second line, 8 × 7 = 56; in each subsequent line, the left side of the equation is multiplied by a number that decreases by 1 from the previous line.
Applying this pattern, the final answer is 9.
Not So Fast!
There might be a way to critique the above solutions using logic, which may validate what the best solution is.
For instance, were we to observe the sequence in its entirety, we may notice a pattern on the left side of each equation, from top to bottom: 9, 8, 7, 6 … etc. Each subsequent line decreases by 1. From this, we may notice that there is a “missing” line in the sequence: 4.
Were we to fill in that missing line and apply it to the first solution (a × (a – 1) = b), the final solution would remain the same as before: 3 × 2 = 6.
Yet, interestingly, when we insert that missing line into the two “secret” solution sequences, the answer changes, and as it turns out, is the same as our initial solution: 6. This goes a long way in validating the correctness of that first answer. Wouldn’t you agree?
Now are you ready for a surprise bonus question? Try this one on for size!
Can You Solve the Sequence? There Are 2 Solutions (But You’ll Need an IQ of 130+)
Now some of you skeptics might be looking at this mind bender and thinking: “The answer is 19. It makes no difference that two of the previous equations are wrong.” And as far as we’re concerned, you’re correct in thinking that! If you declare 19 to be your answer, we’ll take it. That’s thinking outside of the box!
But for those who really want to test their metal, this mind bender has a few more tricks up its sleeve, as you’ll see.
There are a couple of hidden patterns in the sequence of equations that ties it all together and gives you the real answer. But you’ll have to figure out that pattern and solve it mathematically. Are you up for it?
Let’s take another look:
Take a moment to consider what patterns are at work. Hint: there are two possible solutions (besides the aforementioned trick answer of 19). Try to come up with both solutions before you scroll down to see the answers below.
Solution 1 Answer:
The first equation makes plain sense: 1 + 4 = 5 , naturally. But that’s where the logic seems to end. The second and third equations, 2 + 5 = 12 and 3 + 6 = 21, do not equate unless there is a larger pattern or hidden rule that we aren’t given in the sequence. If we can determine what that is, we may be able to solve the last equation.
Add the left side of any given equation to the answer of the previous equation (as per the illustration below). In the case of the first equation, there is no previous equation, and so you would add zero to the left side of the equation (0 + 1 + 4), which gives you 5. The same pattern works for the second and third equations, and so we know it’s correct. Apply this rule to the last equation and we get the solution.
The answer for Solution 1 is 40.
Following this pattern, we can solve the final equation by adding the previous answer (21) to the left side of the said equation (8 + 11), which gives us 40.
Solution 2 Answer:
You might have thought that that is all there is to it, but in fact, as some of you may have noticed, there is another pattern hidden in the sequence, and it gives a second solution that also works. It uses the first pattern but results in a different, yet perfectly valid, answer.
The Second Pattern:
Observe how the first number from each of the first three equations creates a consecutive sequence: 1, 2, 3; and how the second number from each of the first three equations creates another consecutive sequence: 4, 5, 6.
The last equation may seem to break this consecutive pattern, but in fact, it fits into the sequence if you grant that parts of that sequence exist but are omitted from view, as per the illustration below:
And since the entire sequence has now changed—including the second-last equation in particular—and supposing we use the same rule as Solution 1 to solve the last equation, we will get a different answer, as you can see from the illustration below:
And the answer for Solution 2 is 96!
Photo Credit: The Epoch Times