There are some tips to find the notes and intervals on the neck fast.
There is a quick way to learn all the notes on the mandolin neck. Therefore we work with the root notes of the g-string and d-string. These can be divided into four groups:
The groups are based on the dots, the circles in the diagram. These dots will be (almost) always at the same place at many mandolins and may occur both at the front and at the side of the neck as dots or other decorative element.
The notes of group 1 are situated on the first three dots. These notes are the b, c, d, f, g en a. Note: the note b is located in between the first and second dot. Apart from note e you can now find all the natural notes at a place on the mandolin neck. The notes of group 2 are situated on the last separate dot. The notes of group 3 are situated one square before the last separate dot. The notes of group 4 are situated one square before the first dot.
When you know the natural notes on the g-string and the d-string you can octave up to find the notes on the a-string and the e-string. For that you slide two squares to the left (in the direction of the head), and two strings upward (in the direction of the e-string).
On the twelfth fret a double dot is drawn. The notes in these squares are the same notes as the loose strings. If you look further you see - depending on the amount of frets on your mandolin - again a number of dots. These dots are on the same spot as the first part of the mandolin. Thus the neck starts from the twelfth fret again.
This chapter explains that you can lower and raise notes and intervals through respectively the flat symbol (b) and the sharp symbol (#). In addition to the illustrated lowered and raised notes and intervals, there are other, less common lowered and raised intervals.
This way you can raise the note b to the note b#, increase the note e to the note e#, lower the note c to the note cb and lower note f to the note fb. These sound enharmonically the same as respectively the notes c, f, b and e.
You can raise the interval 3 to the interval #3, raise the interval 7 to the interval #7, lower the interval 4 to b4 and lower the interval 1 to the interval b1. These sound enharmonically the same as respectively the intervals 4, 1, 3 and 7.
Additionally, you can also double raise and double lower notes. This is explained in chapter 0.4.
To octave works the same for intervals. Again you can both octave up and down.
It's recommended to also learn the intervals 'under' the 1, because melodies are not only played above the root note (1).
When you play the intervals 1, 2, 3, 4, 5, 6 and 7 in succession you hear the (pure) major scale, the Ionian scale. This scale probably sounds familiar. When you remember how you play this scale, you can find all natural numbers (on at least one place); this way only the lowering and rising remain.
Basically, you can play all scales and other melodic music theory when you know all intervals and notes in one place. However, it's better for the overlooking of the neck if you know the notes and intervals in as many places as possible. Later, this allows you to quickly switch between different scales and thereby play faster and all across the neck.