Lair of The Tark
Apprentice had a very simple system in which challenges were resolved.  The first step compared the relevant attribute to the task at hand to see if a roll was even required.  If not the player rolled a d6 to see if they could exceed the challenge number and move on or used a hand waving mechanism, "casting a spell", in order to bypass the challenge and continue with the game.

Gnome will use a similar system of roll and compare but the character will not have a hand waving mechanism nor a base number with which to bypass challenges completely.   Instead a gnome has a higher or lower base die with which to complete the challenge.  Higher dice represent a better base capability in that attribute while lower represented a weaker capability.

Since Gnome does not have a hand waving mechanism to make up for the lack of a way to boost ones stats a system had to be devised that could both give a character a measure of mechanical personality to go along wit the characters history and attitude. 

The system revolves around a series of Traits and Failings.  Traits are specific circumstances and actions that the gnome is exceptional at that gives a bonus dice with which they can roll.  This increases both the minimum and maximum number they can roll to defeat a challenge and move on.  Failings represent tasks and circumstances that the character cannot handle to the full potential of their attribute and have a penalty die to roll in addition to their attribute dice.  Now unlike apprentice where characters were limited by what stats they could derive from the points they spread in character creation, it is dice that determine a gnomes stats.  They are given a spread of dice that can be assigned to an attribute. Depending on the spread that a player chooses determines the number and ratio of Traits/Failings a character has.  Naturally characters who choose a high dice spread are going to have almost no traits but plenty of failings in exchange for overall power.  While those who choose a weak spread will have lots of options for traits while few failings to worry over.  The following table indicates the varieties of spreads available to characters in Gnome:
1d12,1d6,1d6,1d4||5 Traits, 1Failings

1d12,1d8,1d6,1d4||4 Traits, 1 Failings

1d12,1d10,1d6,1d4||4 Traits, 2 Failings

1d12,1d10,1d8,1d4||3 Traits, 2 Failings

1d12,1d10,1d8,1d6||3 Traits, 3 Failings

1d12,1d12,1d8,1d6||2 Traits, 3 Failings

1d12,1d12,1d10,1d6||2 Traits, 4 Failings

1d12,1d12,1d10,1d8||1 Traits, 4 Failings

1d12,1d12,1d10,1d10||4 Failings

1d12,1d12,1d12,1d10||5 Failings

1d12,1d12,1d12,1d12||6 Failings

Now each trait and failing selected by the player will have a rating assigned to them (1d4-1d10)  this rating determines the die rolled in addition to the attribute that will contribute or cause detriment to the roll.  This rating is determined by the specificness and rarity of the challenge or circumstance that comes up. Broader situations that can come up often are rated the lowest while specific and rare situations are given the highest rating.

For example if a player selects "good at fighting" for a trait then they would receive a very low rating as they would receive the bonus from that trait every time combat ensues.  However if the character selects "good at fighting rats while barehanded" then the player can expect to reap a much greater benefit from the specialization of the trait.  Now while there is always some concern considering min/maxing of traits and failings these aspects of the character are always chosen before the game begins and thus unless a character has specific knowledge as to the nature of the challenges and situations a GM has planned for the character such ideas are moot and a character who over specializes to gain the maximum numbers can easily find themselves with a pile of useless traits.  Traits and Failings are there to give mechanical personality to a character and give them a flavor to help them specialize and find a niche within the group.  Those that choose high numbers of broad traits will find that although they rarely roll low numbers they are really not much better off then a character who has opted to not bother with traits completely in favor of having a good number of high base attributes.

Gnome Attributes:
Characters in Gnome have four basic attributes from which there challenges are rolled.  These attributes are Athleticism, Cleverness, Toughness, and Shrewdness.

Athleticism represents a gnomes physical prowess.  The typical gnome can lift around five to seven times their own body and move at about forty body lengths a second.  So compared to humans gnomes are extremely athletic for their size.  While they can't ever hope to lift a human or outrun a cheetah it's not a question of capability but relativity of size.  A gnome the size of a human would mop the floor of any gold medalist in track and field.  That being said a group of gnomes or even a single gnome can take on creatures much bigger then themselves when necessary.   Athleticism is the dice rolled whenever a Gnome is actually physically challenged either through combat or physical exertion.

Cleverness represents a gnomes problem solving capability and knowledge base.  While it's easy to say that a Dweller gnome with access to human books and the ability to read human language is smarter then a Wild Gnome without access to either they can both be equally clever in their own ways.  Gnomes tend to think extremely quick thoughts compared to humans as a result of their drastically different metabolisms and short lives.  As such compared to a human a gnome might seem very quick witted. 

Toughness represents a gnomes physical fortitude and ability to handle tough situations.  Gnomes are by nature very tough nuts to crack.  Their stout bodies and sometimes cheery look hides thick cords of muscle and flexible, tough cartilage (gnomes do not have a true skeleton). Toughness also represents a gnomes psychological and mental fortitude.  Life is tough when it seems like the world considers you an interesting curiosity to capture and study or chew and digest.  It's easy to think that some gnomes are quite depressed and suicide a constant.  However gnomes are nothing else if not stubborn and willing to press on in even the most hopeless situations.  Toughness is the roll that comes up when a gnome faces a dire physical or mental challenge that can cause them harm.  So far I haven't really figured out yet how I want to derive damage (mental and physical) from these stats though I think some Combination of Toughness + Another attribute to build a sort of HP base (Mental, Physical, Emotional) or perhaps a death spiral system based on this combination.  We'll see.  Opinions are welcome here.

Shrewdness represents a characters instinctual ability or natural caution.  Shrewd gnomes can sense storms by the sweat in their palms, feel food in the way local birds are circling above, or no that humans are close by the shaking in the ground.  Shrewd gnomes are natural survivors and can sense things in another gnomes speech that gives way motives and treachery the other gnome might think that they're hiding.  Shrewdness is the attribute rolled when instinct, common sense, and wisdom will reveal things to the gnome that his cleverness might keep hidden.