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The interesting thing about this problem is that most people when they first encounter it put an artificial limit on the possible solutions they search for. Namely, they think of the nine dots as a square within which the lines must be drawn. This is done quite often and the lesson we learn from it is that we must actively challenge the limits we put on our searching, and actively open our minds to new possibilities.
Following on from considering that the forms of these words other than the singular some notable things can be seen about the occurrence of all the forms:
a day / the day 365 times
days (ayawm) 27 times
two days (yawmein) 3 times
their day (yawmahum) 5 times
your day (yawmakum) 5 times
that day (yawma-ithin) 70 times
The total number of plural form occurrences is 30 (27+3) which is the closest round number to the number of days in a lunar month (29.53 days) and the closest round number to the average number of days in the solar calendar - (30.41 days). Another type of month might be noted here - the time of a Sidereal lunar month. This is the time it takes for the moon to rotate around the earth as measured against the fixed stars. A normal lunar month between full moon and full moon is a rotation of the moon around the earth as measured against the position of the sun.
The length of a sidereal lunar month is 27.321662 days.
The number of plural form occurrences of the word day (not including the dual form) is 27 - again the closest round number to a recognised astronomical period.
The other forms of the occurrences of the word “day” can be grouped into 2 types - merged with words indicating possession i.e. your day / their day and merged with the word meaning a particular one i.e. “that day” the ratio of these forms is 7 (70 / 5+5 ) which is the number of days in a week.
To complete the picture these should all tie together in some way so that each number has its own particular purpose. This is found very easily by taking one number from each form other than the singular: We take “their day”, “that day” and “two days”) we have 5 x (70+3) = 365. We now have again the number of days in a year!
The word month occurs in various forms the following number of times :
a month / the month 12 times
the months (ash-shuhur) once
months (ash-haar) 6 times
two months (shahrein) twice
We can see that the total number of plural form occurrences is 9 (1 + 2 + 3). This is a well known time period for the length of a human pregnancy. (If you think about it no other number of months has a similarly significant meaning). It might also be noted that the result of multiplying the various forms of plurals we reach again 12 months for the year (1 x 2 x 6 = 12).