Odd-Even Mathematical Miracle
Introduction to the “miracle”
There is a widely spread claim, circulating on the web at least since 1998, that there is a “mathematical miracle” in the Qur’an concerning the sums of the surah numbers, and the sums of the ayats, or verses. It is sometimes called a binary, odd-even, or checksum miracle. It supposedly consists of two apparently remarkable coincidences. If you look at the Excel spreadsheet after you’ve read this article, you can see them for yourself and verify what is said.
If you add up all the surah numbers from 1 to 114 (1 + 2 + 3 + … + 112 + 113 + 114), the total is 6555. If you add up the number of ayats for each surah in the Qur’an (7 + 286 + 200 + … + 4 + 5 + 6), the total is 6236.
Now, for each surah, you can add its surah number to the number of ayats it has (e.g. 1 + 7 = 8 for the first surah, 2 + 286 = 288 for the next surah), and we can call the result its “s+a number”.
If you add up all the odd s+a numbers the result is 6555, which as we saw is the sum of the surah numbers in the Qur’an. If you add up all the even s+a numbers, the result is 6236, which is the sum of all the ayats in the Qur’an. Apparently we have a pair of amazing coincidences. Someone might imagine the odds are thousands to one, but are they?
In this article, we will show that these properties of the numbering of the Qur’an are in fact something more simple than it at first seems, and why it is perfectly plausible that it happened by chance, without any deliberate intention, whether divine or human.
1 coincidence, not 2
Before beginning, it’s worth mentioning that the number of ayats into which the Qur’an is divided was not part of Muhammad's "revelation", but rather multiple numbering systems were used (6000, 6204 etc.), perhaps all with an identical Qur'anic text (that's a separate question), and the 6236 divisions of the Kufah school simply became most popular.
The first thing to notice is that added together, the sum of the even s+a numbers plus the sum of the odd s+a numbers must equal the sum of all the sura numbers plus all the ayats in the Quran (since every sura belongs to one of the two groups). Thus if one group = the sum of the suras, it just follows trivially that the other group must = the sum of the ayats. One half of the “miracle” automatically implies the other. We can further deconstruct things from another angle. We can simply state the “miracle” as follows:
1. Total s+a (odd) = total surah numbers
6555 = 6555
2. Total s+a (even) = total ayats
6236 = 6236
Each sura belongs either to the odd s+a group or the even s+a group. If we subtract all the suras numbers belonging to the odd s+a group from both sides of the first equation, we can see that it becomes:
Total ayats in the odd s+a group = total sura numbers in the even s+a group
If we then subtract all the ayats belonging to surahs in the even s+a group from both sides of the second equation we can see that it becomes:
Total surah numbers in the even s+a group = total ayats in the odd s+a group
Swap the sides of this equation round and you’ll see that it is identical to the other equation. To illustrate visually:
Thus both apparent coincidences in reality simplify to a single one. They are not two independant coincidences. They both follow trivially from a single more mundane-sounding coincidence: Total ayats in the odd s+a group = total surah numbers in the even s+a group. Specifically, that number is 3303. And no, 3303 is not divisible by 19 in case you were wondering.
It’s worth noticing that you get the two apparent coincidences simply from the fact that total ayats in one group of suras = total sura numbers for the rest of the suras. So long as this is true it makes no difference if you define the two groups of suras based on odd and even numbers or any other selection criteria.
It is sometimes claimed that the total number and distribution of ayats has to be exactly as it is for the “miracle” to work, and thus it supposedly has a useful function as a sort of checksum against change (in the numbering at least). However, there are many ways that the distribution of ayat numbers could be different without affecting this property and which you can easily verify with the spreadsheet. For example, you could add/subtract any multiple of 2 to the number of ayats of any sura in the even s+a group. Or for 2 even numbered suras, you could swap their number of ayats if both are even or both are odd. The same for 2 odd numbered surahs.
If it was by design rather than coincidence
Incidentally, while this property was surely an unintentional coincidence rather than deliberately implemented by an obsessive numerologist, if the two numbers had not matched (3303 = 3303 as it happens), but rather there had been a gap, it would actually have been very easy to intentionally alter one of the existing numbering systems to close the gap so that there is a match (which is the cause of the 2 interesting properties). As explained in the next section, the gap would probably be relatively small to start with. The following process just involves changing how some of the surahs are divided up into ayats. It does not involve changing the text itself or its order.
If the gap was an even number, you would simply add an even number of ayats to some suras in one group (either the odd or even sura+ayat group), and/or subtract an even number of ayats to suras in the other group until the gap is closed. You would spread the required change over many suras. This needn’t mean changing the text or its order, but rather just the number of ayats into which it is divided, which is easy since 1 ayah does not generally = 1 sentence.
If the gap was an odd number, you would first do an additional step to make the gap even: You would add/subtract one ayah (by changing the divisions, not the text itself) to a sura whose sura number and number of ayats were both odd. You would then add/subtract one ayah to a sura whose sura number was even but whose number of ayats was odd. The gap will now be even because you have changed the total of sura numbers in the even s+a group by an odd number, and you have changed the total ayats in the odd s+a group by zero or an even number.
How remarkable is this coincidence?
Even by the most basic consideration at the start of the previous section we saw that one half of the “miracle” automatically implies the other. We further saw by putting it another way that its proponents unwittingly count the same quantity within both sides of each equation so it is in fact just two versions of the same equation, the same single coincidence. And as we shall see, due to various factors, the likelihood of its occurance by chance is not so low that anyone should be impressed by it and proclaim it a miracle.
The sum of the ayats (which range from 3 to 286, skewed such that the higher numbers are less frequent) is approximately the same as the sum of the surah numbers (which range from 1 to 114, uniformly distributed), 6236 and 6555 respectively. It is not particularly remarkable that you can use some criteria to select approximately half the surahs (as this process does – exactly half as it happens), and find that the sum of those surah numbers = the sum of the other half’s ayats (3303 is approximately half of 6236 or 6555).
Even if your selection turns out to be weighted toward the higher numbered surahs, then the ayats of the other surahs will similarly be weighted toward the higher numbers (since the surahs tend to be ordered such that as the surah number increases, the number of ayats per surah decreases). So there is a rough correlation – they are not independant variables. Almost whatever your selection of half the surahs, the sum of their numbers will roughly correlate with the sum of the other surahs' ayats.
We should also bear in mind that for each way of dividing the surahs into two halves, you have two chances to find a match: your “odd” group might = the sum of all surahs and the “even” group = the sum of all ayats, or alternatively, your “odd” group might = the sum of all ayats and the “even” group = the sum of all surahs.
If the selection criteria that is used hadn’t produced a coincidence, there are many other options for selecting half the surahs and people could have tried them (for example odd numbered surahs, surahs with an odd number of ayats, or words, or letters etc.).
Using computer simulations with random numbers and a similar distribution of ayats as we have in the Qur’an, you'll see that the odds of finding a match after randomly selecting half the surahs are approximately 1 in 170. See the footnotes for the vbscript used. If there is a 169 in 170 chance that a selection criteria will not give a match, then 1 – (169/170)^n gives the probability that you will find a match with n attempts using random selection criteria. For example, there is a 0.057 probablity (1 in 17 chance) of getting a match trying 10 selection criteria. Of course, if you succeed that does not mean that there is a 16 in 17 chance that you have found a miracle. Otherwise every unlikely event would be a miracle. Unlikely things happen all the time.
We must consider that huge amounts of man-hours have been spent looking for numerical patterns in religious books, so try hundreds or thousands of possible patterns and coincidences, including this kind, and it is likely that you will find some. If the numbering had been very different, then obsessive numerologists would have found some different numerical “miracles” instead.
- Mathematical Miracles - A hub page that leads to other articles related to Mathematical Miracles
- So-called Binary Checksum Miracle - XLS file