Geoff.... how does one read, feel, calculate the effect of two breaks on a level green... and even a third break when the ball speed is decaying rapidly and goes wobbly as it nears the cup? Then there is the overall slope of the green which adds another variable.
To simplify the complexity, perhaps you could assume a 24 ft. putt with an initial 12 ft. break and then another 12 ft. opposite break, or something like that. Also comment on different break point reversal ratios.
Could you also relate the path(s) of the putt to that oft-used graph showing the skid/roll/decay of the ball at various speeds? And what about different green speeds? Thanks.
First, the brain naturally predicts these breaks if you "run the movie of your actual ball pace over the exact contour all the way to the hole". So you know the pace extremely well near the hole, since these are always identical at the end of the putt, and you work the pace backwards. With this total-curve sense of ball pace, you then predict the curve's shape, which can also proceed from the hole in reverse back to the ball. This is naturally what brains do, just applied to the ball-on-surface-with-exact-pace pattern and situation.
Second, you can use math to segment the different slopes. Slopes on greens aren't really separate and distinct, but blur from Slope% X to Slope% Y smoothly thru the intervening slopes. The "segmenting" is dividing the slopes into "basically large flat areas," or one Slope% that stays the same for a while before changing into another Slope%. This ignores the transitioning area between segments, but that's not too far off.
Assume three different Slope%'s and their flat areas on the way to the hole -- the first Slope% being 2% with the fall line to the left of your ball so the break is right to left; the second flat area has a Slope% of 3% also right to left; the third flat area has a Slope% of 1.5% also right to left. This is more or less going from usual slope over a bit of a hump or cone projecting into your path from the right with greater slope and then off that onto a milder slope to the hole.
The break over these three slopes can be "summed" but each slope contributes break according to the TIME the ball spends on that slope, not its extent or size. Hence, the TIME is a combination of the SIZE or LENGTH across the slope for the ball path and the ball's average pace or VELOCITY over that area.
Over the first slope nearest the ball, the ball starts with maximum velocity and slows down at some rate (which is more pronounced if the ball is skidding at rhe beginning), then over the second area the ball is slowing at some pretty steady rate depending upon green speed, and then at the end the ball's arrival at the hole is the same as always, finishing up with a decelerating pace so the ball makes it to the hole and goes in nicely or rolls only a short distance past the hole and stops.
Assume each flat area is EQUAL SIZE -- perhaps 10 feet across each flat area for a total of 30 feet. This corresponds on usual green speed to a starting maximum velocity of about 300 inches per second (57 revolutions per second). Across the first slope this might slow fairly suddenly due to skidding and green friction over the first few feet then settle into a usual deceleration, exiting the area at perhaps 160 inches per second (30 revolutions per second). Then across the second flat area the ball continues to decelerate steadily, perhaps to 70 inches per second (13 revolutions per second). Then across the third segment, the ball decelerates down to say 3 revolutions per second at the front lip and zero just a few rolls past the cup.
The AVERAGE VELOCITY across each 10-foot patch is then 300 + 160 / 2 = 230 inches per second on the 1st patch; 160 + 70 / 2 = 115 inches per second on the 2nd patch; and 70 + 3 / 2 = 36.5 inches per second over the final patch.
The TIME the ball spends on each 10-foot or 120-inch patch is simply Length / Average Velocity, so the three patches have this TIME:
1st patch 120 / 230 = 0.52 seconds
2nd patch 120 / 115 = 1.04 seconds
3rd patch 120 / 36.5 = 3.29 seconds
So the total putt TIME is 4.85 seconds. The PROPORTIONATE TIMES for each segment are:
1st segment 10.7%
2nd segment 21.4%
3rd segment 67.8%
With the proportionate contributions of the TIME, you can get an "overall, average Slope%" for the entire putt, ignoring the transitional areas.
So in our case, the average slope% is just the sum of the weighted segments:
2% x 0.107 + 3% x 0.214 + 1.5% x 0.687 = 0.214 + 0.642 + 1.031 = 1.857%, or 1.9%.
So it's roughly the equivalent of a 30 foot putt across one flat area of 1.9% slope. That's not much different than a 2% Slope that breaks 1" per foot, so the break is about 30"' with a target aim spot 30" up the fall line from the center of the cup.
If the breaks go in different directions, the first slope is positive and all others breaking that same way are positive, and opposite-breaking slopes are negative. Find the TIME per segment and the sum the proportionate contributions.
It's helpful to notice that putts that break different directions basically have less total break than the usual putts, and sometimes the break one way just about cancels out the other break, so the putt is pretty straight.
Another fact to notice -- after years of experience and observation, of course -- is that most "multiple breaking putts" break over different slopes that all break the same general way -- left to right or right to left. That is, your ball is on the same side of the fall lines over each separate flat-but-tilted area -- balls to the right of fall lines aiming uphill break to the left downhill, and balls to the left of fall lines break downhill to the right. Considering two such slopes, the one closest to / at the hole is the decisive slope, and the one to care most about. These final slopes come in two flavors: steeper than the first, or less steep than the first. If the final slope is steeper, play more break, and if not, play less break.
These two slopes also come in two other flavors: last slope uphill or last slope downhill. If uphill, you have a green light for pace, since the increasing slope when the ball is slowing really acts in combination to decelerate the ball. If the second slope is downhill, or more downhill than the first slope, be careful but also realize that the ball will be slowing onto the second slope so try to avoid being SO CAREFUL that the ball gets hung up on the second slope and never makes it to the hole. Whenever the final slope is downhill for ANY putt, be sure to look carefully at this final slope from behind the hole, reading the putt into the cup down that slope with the fall line of that slope clearly in mind.
Putts that actually CHANGE directions from breaking left to right and now breaking right to left MUST HAVE CROSSED OVER A FALL LINE. This tells us something important: the break direction does not change unless the putt crosses over a fall line, so if you don't see one, the putt is probably not a double breaker.
Generally, it is more or less the case that putts spend about 2/3rds the total time over the last half of the putt, so that helps. If a double breaking putt travels over 3% then 2% with both breaking right to left, and both about half the putt, the average or overall slope will be a lot closer to 2% than to 3%. The actual math using this rough idea is 2 x 0.67 + 3 x 0.33 = 2.33%. This rule of thumb is not exact, but is reasonable.
There are obviously some common situations that are tough to read, involving more than the simple break over the same flat-but-tilted area, such as putting off one slope up onto a ridge or bowl, and putting up along a "river bed" with the hole slightly off the river on a side slope, but that's why I teach lessons and clinics.
You can always more or less ballpark a mathematical aim spot on the fall line by averaging the slopes. A really rough way to get started is to notice that almost all greens slope from front up to back, so that amateurs don't have to play to away-sloping greens. So if you go to the front of the green where the fringe is lowest and look across to the back of the green at the point where the fringe is highest, and estimate the difference by standing your putter shaft up from the low fringe and sighting level across the vertical shaft to the back fringe while pretending to pat your hand on the back fringe, and noticing how far up the shaft in feet the hand is patting, you can then ballpark the green's total overall average slope.
It's very common to find that greens have a general or overall slope of about 2%. This seems to be something of a default slope since 1% may not drain enough and flattish greens are hard for amateurs to read, while 3-4% slopes may slough the irrigation or rain off the surface too quickly without benefitting the roots and also present difficult putts for amateurs. So the 2% average slope is sort of a "design sweetspot".
All you need to know to finish this is the distance to the back fringe to get the percent slope (rise on putter over run from front fringe to back fringe). Conveniently, many many greens are 33 yards in depth, so that is 100 feet. In that case, the green's overall slope is whatever number of feet your hand seems to be raised above the low fringe while pretending to pat the high fringe on the other side of the green. If the hand is at the bottom of the grip material, that is probably 2' high, so the green has an average slope of 2%. If the run from your low point to the high point is only 50', and your hand is 1' high when patting, the slope is just twice the hand height, so agan 2% average slope. Divide the hand height in feet by the distance from fringe to fringe in feet to get a sense of the overall slope percent.
This overall average slope often helps when planning a long lag acros some complications, but in general across an average slope of "about 2%" or whatever your estimate. For example, a long lag of 55 feet across some complications but nothing crazily different from the overall average slope has a "pretty close" ballpark aim of 55" up the fall line at the hole when the average slope is 2%. That's about 1 and 1/2 putter lengths up the fall line from the cup. If you aimed there as a start, and then assessed whether the ball would stay on the high side all the way to the cup if you putted normal pace, then that target and start line is very likely agood one.
Remember, these multiple breaks are not likely to present inside 10 feet, so when these multiple breaking putts do present, they are most likely to be long lag putts that have priority on getting the first putt within 2 feet or so of the hole for a safe two-putt. That's where the lag X comes into play.
To the extent you are asking how to read these putts EXACTLY, well, to hell with that because it's pointless and stupid and not going to happen and too difficult in math and unwise to think you can and will make it harder to do as good a job as you are able to do. Just lag these close.