Now most people including some media agree that it has to be one of two solutions: either 40 or 96. I don’t agree with either solution, because for both variants you have to assume that part of the equation is written in invisible ink so to speak. Instead I have a better solution. One, in which 5 + 2 = 12 is actually true. And no, I’m not nuts. But let’s take a look at the two most common solutions first.
In order to come to 40 as the solution, people add the result of the previous line to the current line:
1 + 4 = 5
5 + 2 + 5 = 12
12 + 3 + 6 = 21
21 + 8 + 11 = 40
Now this is a beautiful pattern which of course makes sense once you’ve added the previously invisible numbers. But, apologies for being a stickler for the rules, it has one crucial flaw: it just doesn’t stand there. The maths puzzle doesn’t say 5 + 2 + 5 = 12, it says 2 + 5 = 12.
Similarly, to reach 96 as a solution, first you have to double the first number, and then assume that the second number is multiplied by it:
1 + (1 x 4) = 5
2 + (2 x 5) = 12
3 + (3 x 6) = 21
8 + (8 x 11) = 96
Now, as already mentioned, I’m a self-confessed stickler for the rules and I don’t like solutions that force me to assume that parts of the equation have been written in invisible ink. When I came accross the puzzle a couple of hours ago, I naturally assumed that there has to be a way these equations make sense in exactly the way they are written, and since I stubbornly clinged to that assumption, I had to let another assumption go: that 2 + 2 always makes 4. Or that 2 + 5 always makes 7. Because of course it doesn’t. There are other numeral systems, and the decimal numeral system is only one among many. Or have your Maths teachers never tortured you with binary numbers for instance?
So my first thought was: binary. But of course you can see at a glance that that doesn’t work, 12 can’t be binary. The binary number system has only two numerals, 0 and 1.
But nevertheless, once your eyes are opened for the possibility that this might be formulated in another numeral system than the one we’re used to, you’ll be looking at the line
2 + 5 = 12
from a completely different angle. And actually that makes the first part of the riddle quite easy. Just give it a try. It’s obviously not decimal, and it’s obviously not binary, so in what kind of numeral system could this be true? In the decimal system, obviously, 2+5 makes 7. Assuming the right part of the equation is written down in another numeral system, then 12-in-that-other-numeral-system has to be the equivalent of 7-in-the-decimal system.
If you want to find it out by yourself, just take your time. And don’t scroll down.
Now, you may not know the expression, but the numeral system which is based on the number five is actually called quinary. Just like the decimal system is based on the number ten and the binary system on the number two, the quinary system is based on the number five.
In the quinary system you have five numerals: 0, 1, 2, 3, and 4.
When your counting reaches the number five, you write that as 10. In the decimal system 10 means 110 plus 01. Just like 11 in the decimal system means 110 plus 11. And so on. Now we’re in the quinary system, and here 10 means 15 plus 01. Which is five. Just imagine you’re counting with your hands if this way of thinking is unfamiliar to you. When you have a two digit number in the quinary system, then with the first digit you count the number of hands so to speak.
Now coming back to our equation:
2 + 5 = 12
is mathematically 100% true, without any addenda, if you assume that on the right part of the equation we have silently shifted to the quinary system. 12 in the quinary system corresponds to 15 plus 21 in the decimal system, which is exactly what is written on the left side of our equation. Nice, isn’t it? And, at least for sticklers like me, the nicest part is we don’t have to bend any rules or assume that parts of the equation have been written in invisible ink.
Now let’s take a look at the third line:
3 + 6 = 21
Of course the first intuition is to apply the method of the previous line, but sadistically that doesn’t work. 21 in the quinary system is equivalent to 11 in the decimal system, not to 9. sigh So we have to think again. In which system could 3 + 6 = 21 be true?
We know that 3 + 6 equals 9 in the decimal system, so what we need to figure out is this
2x + 11 = 9
We can simply transform this into
x = (9-1) / 2 = 4
And voilá, there is our answer: we’re now in the numeral system which is based on the number 4 – the quaternary system. Let’s check:
In the quaternary system 21 is 2 times the base number, which is 4, plus 1 times 1:
3 + 6 = 24 + 11 = 9.
That’s it.
So obviously on the right side of our equations we are counting down the base number of our numeral systems. Let’s take a look at it anew. We silently assumed that the first line entirely belongs to the decimal system. That’s possible. But the right part of it could also belong to the senary system, which is based on the number 6. Or any other system whose base number is higher than 6. But since 5 is the highest numeral in the first line, the base number has to be at least 6.
The right part of the equation in the second line belongs to the quinary system, based on the number 5. And the right part of the equation in the third line is formulated according to the quaternary system, based on the number 4.
So what comes more natural, than to assume that the right part of the fourth equation should be expressed in the ternary system, which is based on the number 3?
In the ternary system you have three numerals, 0, 1, and 2. When you count to three, you write it as 10. Which stands for 13 + 01. Let’s just count a little bit. In the left column, we use the ternary system, in the right column we use the decimal system:
1 = 1
2 = 2
10 = 3
11 = 4
12 = 5
20 = 6
21 = 7
22 = 8
100 = 9
Now we almost got it. To express the number 9 we need three digits in the ternary system. The decimal number we want to reach is 19. That is 2*9 + 1.
In ternary numbers 2*9 is 200. Plus 1 is 201.
201 = 19 = 8 + 11.
And that is the answer to our maths puzzle. If we count down the base number of our numeral system on the right side of our equations from 6 to 5 to 4 to 3, the answer to our riddle is
8 + 11 = 201
We can now demystify the entire riddle by making the numeral systems which are used in each line explicit, which we can easily do by adding some brackets and subscripted numbers which indicate the base number of each numeral system*:
(1 + 4)10 = (5)6 (or higher)
(2 + 5)10 = (12)5
(3 + 6)10 = (21)4
(8 + 11)10 = (201)3
Of course only idiots would say that the solution is twohundredandone. But it is definitely 201.
Agree?
