A rune was an enhancement that the summoner provided for their champion before a match on a Field of Justice began to augment the champion's abilities. In lore, they were magical artifacts of immense power.
Runes could only be purchased from the Riot Store with Influence Points. Runes could not be purchased with Riot Points. Each rune granted a small bonus in a specific category, which stacked upon each other to grant larger bonuses.
Glyphs, marks, and seals stacked up to 9 for increased effect, and quintessences stacked up to 3.
Purchasing the maximum number of every type of available tier 3 rune (9 of each glyph, mark and seal, and 3 of each quintessence) would have cost 388,071.
The number of runes a player can have was limited to approximately 720. Note that getting the appropriate maximums of each tier 3 rune came out to 684, however, this amount can be exceeded if you purchased either special event runes or purchased more than the maximum usable amount (i.e. 9 quintessences) of a specific rune before patch 1.59.
Additional Rune Pages may also been bought from the Riot Store with Riot Points or Influence Points, up to a max of 20.
A Thousand years ago, magical artefacts called ‘World Runes’ have been rediscovered. Over the following decades after such event, knowledge of the Runes began to spread as more were unearthed. The world’s brightest minds studied the ancient glyphs, attempting to determine the powers they possessed. Few could even begin to understand the importance of their origins, or the sheer power held within them. Some surmised that the Runes were integral to the creation of Runeterra itself. The first use of these mysterious artifacts proved catastrophic, as they reshaped the landscape of entire nations. Mistrust quickly grew as those who knew of the Runes imagined such “Makers’ Might” being used as a weapon. Conflicts later related to these magical artifacts were known as the Rune Wars.
It is also important to note that some runes were "primary", while others were called "secondary" and provide lesser bonuses. Given a certain kind of rune, it would be primary in either marks, glyphs or seals, and secondary in the other two rune types. So, all kind of runes had one primary type for them and two secondary types. The aura shrouding a rune of a certain effect would denote which rune type it is considered primary - for example, ability power runes would have a blue aura showing that it is Glyphs that specialize in them.
These Marks had the strongest effect among non-Quintessences thus making them primary.
Quintessences had the strongest possible effect from a single rune and they also have couple effects that can't be found among other rune types.
Although Quintessences had the strongest possible effect from a single rune, you may only have had three instead of the nine marks, seals, or glyphs. This means one quintessence should have provided statistics that equaled to or exceeded 3 minor runes in order to be a Primary Quintessence. The following Quintessences had an effect equal to or greater than three primary runes:
The Tier of a rune represented its relative power (i.e. the extent of the bonuses it offers), and was visible in the top-left of the pop-up when mousing over it.
Tier 1 or "Lesser" runes were dark and faded with visible scratches and chips except for Quintessences, which were simply purple.
Tier 2 didn't have prefixes, they were slightly brighter and are not tattered; Quintessences now had gold faces and purple backgrounds. You could buy them starting at level 10.
Tier 3 or "Greater" runes were brightly lit; Quintessences were completely covered in gold. You could buy them starting at level 20.
It is important to note that you could not buy a Tier 3 rune before you reach level 20
Summoners are able to use a rune per level of power in League of Legends, and they keep their Runes in a tome called a Runebook. Before a match begins, a Summoner will be able to choose from up to twenty different rune configurations they have previously set in their Runebook. This allows the Summoner to have quick flexibility in choosing a Runebook configuration best suited for whatever champion they might use in any of Valoran's Fields of Justice.
For the purpose of estimating the runtime of search algorithms in the space of possible rune pages, it may be interesting to enumerate the number of unique possible combinations of runes. Since the position of any particular rune on the rune page does not influence its statistics, the order within the rune page can be disregarded and it only matters how many of which types of runes occur on the rune page. In the mathematical sense, this means we are looking for the number of combinations with repetition (since any type of rune can occur multiple times, in fact as often as there are slots) of length 9 from the sets of marks, seals and glyphs and of length 3 from the set of quintessences, respectively. The problem can be solved with binomial coefficients. Assuming only Tier 3 runes are chosen and no slots may be left blank:
19 different T3 Marks constitute c(19+9-1, 9) = 4,686,825 combinations of marks.
24 different T3 Seals constitute c(24+9-1, 9) = 28,048,800 combinations of seals.
22 different T3 Glyphs constitute c(22+9-1, 9) = 14,307,150 combinations of glyphs.
33 different T3 Quintessences constitute c(33+3-1, 3) = 6,545 combinations of quintessences
The number of unique combinations of entire rune pages is the product of those:
12,309,936,280,199,112,555,000,000 or ~1.23*10^25 or 12.31 septillion.
If an algorithm was to evaluate one billion of these combinations per second, it would still require ~ 390.08 million years to evaluate all of them, making exhaustive approaches practically impossible.
Either preselections of eligible runes or heuristics have to be employed instead.