The Viral 1 + 4 = 5 Puzzle – the Correct Answer Explained

- 4 MINS READ

By Presh Talwalkar.

This riddle was posted to Facebook with the claim that only one in a thousand will figure it out.

1 + 4 = 5
2 + 5 = 12
3 + 6 = 21
8 + 11 = ?

What do you think the answer is?

The problem went viral and generated over 3 million comments with people arguing about the correct answer. What do you think the correct answer is?

Answer To 1 + 4 = 5 Puzzle

A mathematician might take a literal approach.

1 + 4 = 5
2 + 5 = 12
3 + 6 = 21
8 + 11 = ?

The first equation is true, the second and third are false, and the answer to the equation should be 19.

But riddles like this are not about literally interpreting mathematical symbols. They are about identifying a pattern in the set of equations and applying it to the unknown.

The answer that jumped into my mind is to add the first number to the product of the two numbers to get the answer. That is:

a + b means a + ab

This works for the known equations.

1 + 4 means 1 + 1(4) = 5
2 + 5 means 2 + 2(5) = 12
3 + 6 means 3 + 3(6) = 21

Applying the pattern to the last equation gives the answer of 96.

8 + 11 means 8 + 8(11) = 96

The answer of 96 is valid according to this interpretation. Furthermore, many other IQ tests have this sort of pattern where you take two numbers and find a hidden equation that involves simple operations like addition, subtraction, multiplication, division, and exponentiation.

But some people interpreted the problem differently and arrived at a different answer.

Another interpretation: running total

Some people felt the pattern was a running total: add the result in the previous line to the new numbers to get the new answer.

The first line is valid mathematically.

1 + 4 = 5

For the next line, take this result of 5 and add it to the new numbers to get the new answer.

5 + 2 + 5 = 12

Do the same thing for the next line: add the previous line’s result of 12 to the new numbers to get the next result.

12 + 3 + 6 = 21

To solve the puzzle, do the same for the final line.

21 + 8 + 11 = 40

This pattern results in an answer of 40, and many people suggested this answer is more valid.

So what is the answer? Is it 40 or 96? While we cannot know for sure what the puzzle maker had in mind, there is a way to reconcile these two approaches. It turns out the running total can also lead to the answer of 96, if you decide to fill in the pattern a bit more.

Running total missing lines

Suppose the answer in each line is the running sum total of the previous result and the new numbers.

1 + 4 = 5
5 + 2 + 5 = 12
12 + 3 + 6 = 21

Notice that in each line the new numbers are incremented 1 more from the previous line. For example, 1 + 4 is turned into 2 + 5; so both the numbers 1 and 4 are increased by 1. Then 2 + 5 is increased to 3 + 6.

We can continue this pattern, so the next lines would be 4 + 7, then 5 + 8, then 6 + 9, then 7 + 10, and then 8 + 11.

What is the running total when we include these “missing” lines?

1 + 4 = 5
5 + 2 + 5 = 12
12 + 3 + 6 = 21
21 + 4 + 7 = 32
32 + 5 + 8 = 45
45 + 6 + 9 = 60
60 + 7 + 10 = 77
77 + 8 + 11 = 96

We once again get to the answer of 96, which is somewhat surprising!

In fact, the running total with the missing lines generates the same answer, line by line, as the algebraic result from the first approach:

a + b means a + ab = a(1 + b)

How can we see the two approaches are the same? On line 8, there are 7 previous lines. We can make 11 by pairing the first number of each line with the second number of another line: we can pair 1 with 10, 2 with 9, 3 with 8, 4 with 7, 5 with 6, 6 with 5, and 7 with 4. These are 7 pairs of 11. The final line has another 11. This means we need to take 8 and add 11 to it 8 more times, which is 8 + 8(11) = 96.

In general, line n has the equation n + (n + 3), which is equal to the result n + n(n + 3) = n(n + 4).

Let’s prove this formula holds by induction. Assuming the formula is true up to line n, we then consider the next line. In the next line n + 1, we add the numbers (n + 1) + (n + 4). The result in line nis n(n + 4), so when we add (n + 1) + (n + 4) we get:

n(n + 4) + (n + 1) + (n + 4)
n2 + 6n + 5
= (n + 1)(n + 5)
= (n + 1)[(n + 1)) + 4]

And this completes the induction.

Most people believe the answer is either 96–with the equation a + ab–or 40–with a running total. Since the running total can also get to the answer of 96 when extending the pattern to missing lines, many believe that 96 is the answer that makes the most sense.

A Third Interpretation: The Answer is 201.

There was an another obscure answer I found in my research that I omitted. But I am updating this post on January 3, 2018 since several people have emailed me and would like this answer to be represented too.

About 10 people saw this pattern out of 5+ million views this video/post have gotten. So if you found this pattern, then you’re part of a special 1 in 500,000 group!

The idea is to evaluate the sums in base 10, and then convert the answer into descending number bases 6, 5, 4, etc. line by line. So the pattern is:

1 + 4 = 5 (base 10) = 5 (base 6)

2 + 5 = 7 (base 10) = 5×1 + 2 = 12 (base 5)

3 + 6 = 9 (base 10) = 4×2 + 1 = 21 (base 4)

So if we do the same for the last line, we would need to use base 3 to get:

8 + 11 = 19 (base 10) = 9×2 + 3×0 + 1 = 201 (base 3)

So if you got 201, that’s another way people have seen the pattern. It’s a bit more involved to explain what is going on, and you can’t really reconcile the result with the “missing line” interpretation. But several people did see this pattern.


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