When it comes to modern editions of D&D following the D20 system, many player actions are resolved through what is known as an Ability Check. i.e. the player rolls a twenty-sided die adds his ability score modifier (a bonus or penalty) and compares this to a target Difficulty Class (DC).

In earlier editions of the game an Ability Check was popularly carried out by attempting to roll under a characters Ability Score on a d20 but as far as I know this method does not predate (at least) Holmes Basic and possibly later products such as B/X and BECMI D&D.

The earliest attempt I can find in regards to crafting some form of OD&D Ability Check dates back to June 1976 and can be attributed to Wesley D. Ives with his article in Dragon #1. However his method is rather complex and not so elegant. The article is never the less worth a read because he sets out some useful descriptions of what sort of tasks are appropriate for each ability score and this is not "sullied" by any later additions which had already abandoned the OD&D approach of having "baked in" skills requiring a d6 roll and not a d20 roll and this brings me to my main point...

OD&D Ability Checks & Why Rolling Under is Unbalanced

Rolling under an Ability Score is not very balanced in terms of assigning probability of success or failure. For example, lets say a character has a Dexterity Score of 5. If using the "roll under" method they would have a 20% chance of success. This is much more favourable than a 1 in 6 chance of 16.67%. The reason this is important in OD&D is because the general requirement in OD&D when performing common tasks such as forcing open doors, listening and so on is to roll a 1 or 2 on a d6. That's either a 16.67% or 33.33% chance however a character with an average Ability Score of around, 10 or 11 would get a comfortable 45 to 50% chance of success, which is a little too favourable in comparative terms.

As we have now established... In OD&D many things can be boiled down to a chance in 6 with a 2 in 6 chance 33.33% being the normal level of challenge (pits, traps, secret doors etc). So it seems a method using a d6 would be the most appropriate.

Ability Scores are represented in Men & Magic as having low, average and high scores. If you study modifiers associated with Ability Scores I believe they tend to (if not always) range from -1 to +3. So,  by logic alone it would seem appropriate to assign a chance in 6 to an Ability Check based on a characters individual Ability Score, as follows:

18 = 5 in 6 = 83.33%

17 = 4 in 6 = 66.67%

13-16  = 3 in 6 = 50%

9-12 = 2 in 6 = 33.33%

3-8 = 1 in 6 = 16.67%

Scores are reflected as low (3-8), average (9-12) as Men & Magic tells us. High scores are then divided into low-high, average-high, and exceptional (similar to how Greyhawk further breaks down a high Strength Score for purposes of a fighter's strength bonuses).

As you're probably aware I spend a lot of time playing D&D solo and the great thing about having a consistent method such as this is that it allows the solo player to handle non-ordinary tasks without having to "wing it". 

Here then are the complete rules I propose:

HOW TO USE NON-PRIME-REQUISITE ABILITY SCORES

When a player tries to do something out of the ordinary, the table below can be very helpful in deciding fairly whether the character can succeed in the attempt.

Ability Score     Chance in 6     Percentage

18                5 in 6          83.33%

17                4 in 6          66.67%

13-16             3 in 6          50%

9-12              2 in 6          33.33%

3-8               1 in 6          16.67%

USES FOR ATTRIBUTES:

STRENGTH: Used for any tasks requiring extraordinary physical effort.

INTELLIGENCE: Involved in figuring out how to operate mechanical and magical devices, recognizing patterns, understanding cause and effect, identifying types of lairs, learning new languages and skills, and more.

WISDOM: Helps in determining the “correct path” of action, recognizing the function of devices, and similar tasks.

CONSTITUTION: Relates to stamina-related activities like swimming, running, staying awake, enduring hunger, and similar challenges.

DEXTERITY: Pertains to tasks involving manual dexterity, such as handling devices, maintaining balance, climbing, tying or untying knots, and more.

CHARISMA: Affects believability, persuasiveness, follower morale, and related social interactions.

Here is a link to a PDF version of these rules.