Geoff.... How about "putting through the break(s)" typically on a longer putt with several breaks and a puzzling slope of the green?
When I'm totally confused and can't make an educated decision, I just go straight for the hole and hope I don't end up too far past the hole. Sure enough the breaks are smoothed out and the ball ends up nearer the hole, and I have a good read on the return putt...!!!
Surely you must have experienced such a shambolic state, sometimes....!!!
There really isn't anything "puzzling" about reading surfaces of greens if you know how to do it.
First, always imagine the rolling of the ball with the actual delivery pace you will execute the putt with, at each specific velocity at each corresponding point of the total pathway to the hole. Get a delivery pace, learn to see and recognize and imagine it, and then imagine it all the way from ball to hole when reading the surface contour to predict the actual curved breaking pathway the ball will follow when you execute the putt with that same touch.
Second, always read the part of the putt at the end first, where the ball enters the hole, usually the final 2-3 feet of the breaking pathway. Once that is predicted accurately, the golfer should work backwards from this outcome-determining final entry path towards the ball, "filling in the blanks" of how the ball speed from start to finish and the final section will require a specific pathway only. Following this pathway backwards from the final section all the way back to the ball shows the golfer a starting line, which is the final straightness of the breaking path near and then at the ball itself for this predicted breaking pathway.
Third, noticing when the flatness of the surface near the hole changes as the golfer reads backwards from the final section of the breaking pathway at the hole, the golfer simply predicts with accurate imagining how these surface slope changes will require changes in the total breaking pathway, with mild changes in slope steepness being not too hard to predict.
Fourth, with respect to dramatic contour / slope changes encountered working backwards from the outcome-determining pathway into the hole, such as a "shoulder" projecting 10-15 feet into the interior of the green from a fringe-side hump or hillock, the golfer first identifies the exit point off this dramatic feature in the backwards-filling-in imagining, and then "works the feature only" to then predict the entry point onto this feature (hump, hillock, steep tier, swale, etc.), and then simply continues on with the backwards-filling-in imagining on the next slope on the way back to the ball, with the end result that the golfer gets a starting line at the ball, and also knows what needs to happen in crossing over the hump or tier or other dramatic feature in terms of entry point and exit point.
Fifth, with respect to multiple breaks when the surface slope on each section of the putt differs but does not have a dramatic feature in the contour, the fact is that over 90% of these multiple-break situations are breaking in the same direction on each different slope section, with the two main "flavors" being first mild slope at ball then steeper at hole and second steeper slope at ball than at hole -- the first combination breaks more at the hole than might appear at first general taking in of the situation, whereas the second breaks more at the beginning and this slight change in direction early really really matters on long putts as the length of the putt magnifies the early direction change and then at the end the breaking peters out as the ball rolls onto a milder slope.
Sixth, when the multiple-breaking path actually crosses contour that makes the ball break in opposite directions in relation to the baseline straight between ball at address and the hole -- say, for example, a putt starting first to the right off the baseline to the high side and then breaking right to left back to the baseline, but continuing across the baseline to the left which is now the next high side section of the putt and then breaking left to right back towards the hole -- this pattern is only "puzzling" when the difference in slopes is not great, which happens only when both slopes are "not much" and are close to 1%, otherwise the putt really is better characterized as having a dramatic contour feature in the way of the ball at address and the hole, since those dramatic features like tiers and humps and swales can make the ball cross the baseline and curve the opposite way; consequently, on these "puzzling" two-direction pathways, it is reasonable simply to imagine the "net effect" of what would happen if the golfer simply putted straight at the hole for start line to imagine and predict accurately how much to left or right of the hole the golfer believes the multiple-break putt will "net out" when the imagined ball crosses the fall line near the hole with appropriate pace, and then aim the opposite "net" distance to the other side of the fall line at the hole for starting line and send the ball on its way.
Seventh, in the case of multiple slope areas on long lags, the golfer can use a "rule of thumb" formula to get a decent start line with a target spot identified on the high side of the fall line thru the cup, and then using that start line, send the ball with a nice delivery pace to the high side of the fall line like pitching pennies to the base of a low wall up along the fall line, with the ball's actual pathway then curving down towards the hole closing down slowly at the hole from the high side in the final section of the putt; this formula requires generalizing the different slopes into one "average" slope the same way a sandbox sculpture of the actual green contour might get smoothed out with the golfer's hand but smoothed out to the ONE slope steepness that actually is the average of the different slopes; the different slopes count for more or less than the others not really by tape-measure length of SPACE / DISTANCE the ball will spend crossing this or that slope, but according to the amount of TIME it will take the ball to cross this or that slope in proportion to the total time of the putt, so that the first half of a putt that is one half the length is only one-third as important as the second "half" of the putt because the ball crosses the first half of the putt (for distance) in only one-third the total time it takes the ball to reach the hole; the golfer CAN use the different slope areas' lengths, but this is not quite right often, and the golfer needs to get used to thinking more in terms of how fast the ball actually will be rolling on different sections of the putt, and use THAT TIMING as the measurement of how important this slope or that slope really is in the total pathway; once the golfer can average out the different slopes (i.e., first third is 20% of the time and second third is 30% the time and the final third is 50% of the time -- so the 1% slope on the first third is 0.20 x 1%, and the second 2% slope is 0.30 x 2% and the third 3% slope is 0.50 x 3%, so the AVERAGE slope is 0.2% + 0.6% + 1.5% = 2.3%), then the golfer simply multiples that average slope times the number of military-parade paces from ball straight to the hole (where each military step is 30" strides or 2.5' each, every 4 steps being 10'), and this product is how many inches uphill along the fall line thru the cup to aim the start line for the long lag, and then the golfer uses the pace on the putt that sends the ball nicely all the way to the low wall running uphill above the cup along this fall line -- for example, if the above "average" slope 2.3% is on a putt where the distance from ball to hole is 12 paces (a 30-foot lag), then the point to aim at for start line is 2.3 x 30 = 69" up the fall line, or not quite 6 feet (72") or two putter lengths: aim that high and then roll the ball online that far; and a final point is that if the slopes have fall lines that aim to opposite directions or sides of the baseline straight between ball and hole, then these slopes get one sign (+) when all aiming on the same side of the baseline but get the opposite sign (-) when the fall line points to the opposite side of the baseline, when figuring the average slope -- it happens sometimes, for example when the ball has to climb up a hill at first and then past the peak of the hill breaks the other way off the hill to the hole.
Putting Coach and Theorist