For a quick and simple "rule of thumb", subtract one Sign's number from the other - Aries=1, Taurus=2, ... Pisces=12, etc. - and if they differ by 3 or 9 they are a "difficult" match. This is true when matching Sun Signs, and also in most cases when matching the Signs of the Moon and any Planet - except Venus and Jupiter, which rarely indicate a "difficult" influence in a match.
When one partner has the Sun in his birth chart making a "square" aspect (being at a 90-degree angle) to the position of the Sun in the other's birth chart (a difference of 3 or 9 in Sign number), it means compromises will be needed to adjust to very different basic natures and goals. A square is the most difficult aspect, but this does not mean the relationship is "hopeless" by any means - as those overly simplistic Sun Sign compatibility charts (including my own
Love Match List) might lead you to assume.
In relationships, the aspects (certain angles between planets, such as 0, 30, 60, 90, 120, 150, or 180 degrees) between each partner's Moon and Venus to the other's Planets can be a greater influence than aspects between the Sun in their respective birth charts. A Sun/Moon conjunction (one's Sun being very near the same degree in the same Sign as the other's Moon) is one of the best indicators of a happy marriage or fine friendship. A Venus/Mars conjunction is the best indicator of a romantic attraction and a satisfying sexual relationship. "Sextile" aspects of 60 degrees and "trine" aspects of 120 degrees are very good too, though not as strong as the conjunction (0 degrees), when comparing Sun/Moon or Venus/Mars.
But you can't know the positions of the Moon or Venus or Mars unless you have the person's birth chart, so these factors are not apparent from a simple Sun Sign table. That's why you shouldn't make any final judgments after ONLY comparing the Sun Signs.
You really can't be certain you're not very "compatible" just because your Sun and their Sun are NOT located in compatible signs, for many other compatible aspects might offset that one area of incompatibility. You CAN assume that there's a better chance of overall compatibility if your Sun and their Sun ARE located in compatible signs, although examining all the aspects of the Sun and Moon and other Planets might indicate other factors which make you less compatible.
You can use the Sun Sign "love match" tables as simple and somewhat useful general guidelines, but not as the definitive guide to compatibility between two people. For a complete picture, you would need a full compatibility analysis like
the one I offer using the synastry technique.
The Ascendant is also an important indicator of compatibility, as are the Houses of each person's birth chart. To locate these accurately would require knowing the birth time to within a few minutes of accuracy (though a less accurate kind of chart can be created for those who don't know their birth time). Your Ascendant, which indicates your "style", can make "aspects" to the Ascendant, Sun, Moon, and Planets in your partner's chart, some good and some not so good.
When the other person's Sun or Moon or Planets "fall into" certain Houses in YOUR birth chart when THEIR chart is overlaid upon yours, that also indicates the nature of your relationship with that person. Having one person's Sun or Moon or Venus in the other's Seventh House is often a good indicator of "partner" potential in marriage (or business), for example.
There are are other "aspects" between the Sun and Moon and Planets which could indicate "difficult" factors in a relationship, and you won't know these just by looking at a "Sign Compatibility" table. If you have, for example, Sun in Aries and had some difficulty relating to a few Sun in Cancer people, and then you decided it just isn't worth trying "a Cancer" again, be aware that it MAY NOT have been the Aries/Cancer Sun mismatch that was causing the problems.
Those partners might have had their Mars or Uranus conjunct your Moon and seemed irritating, or had their Saturn conjunct your Moon or Venus or Sun and seemed to limit or burden or try to "control" you somehow, or perhaps you and they had several other "difficult" aspect combinations of which you were unaware.
So don't arbitrarily "write off" a certain Sun Sign because you had difficulty with a few people born when the Sun was in that Sign. The Sun might not have been to blame at all!
Compatibility analysis is not a simple process based only on the Sun Sign of the two partners. To arbitrarily "screen out" a potential lover based only on his or her Sun Sign being "incompatible" with your Sun Sign might make you miss out on a potentially great relationship!
What if that person with the so-called "incompatible" Sun Sign happened to have his or her Moon in same Sign as your Sun, or actually conjunct your Sun, and also had his or her Mars conjunct your Venus? Regardless of any "Sun Sign" mismatch, you would be well advised to give this hot prospect a fair chance!
Really, the only way to get an accurate assessment of the potential of a relationship using Astrology is to get the month/day/year/birthplace (and hour and minute if possible) of your potential partner and have a "relationship chart" (either a "synastry" or "composite" chart) created and interpreted by a competent astrologer.
Because two charts must be cast and so many aspects between them must be compared, these take a long time for an astrologer to do by hand and thus can be expensive. Being a "lazy" Taurus, I have programmed my computer to use astrological "synastry" techniques to produce a very detailed and insightful
Compatibility Analysis of 10-15 typed pages, which I sell online for only $30 USD. If you can supply birth times for yourself and your partner, it will also include my personal "Mating Rating" with scores from 1% to 99% on each of seven relationship Factors.