Have you ever noticed how love is a little like a path on a grid? You think you can walk from point A to point B in a straight line, but suddenly you are zigzagging, measuring, second-guessing. And then, just when you think you have the perfect order of distances, someone hands you a list and says, 'Make it happen.' Do you sigh? Do you laugh? Do you reach for your heels or your calculator? In the city of geometry, darling, both are useful.
Here’s our little romantic mishap: five dots are standing on a 7×7 grid at these coordinates — A = (5,0), B = (6,3), C = (1,3), D = (6,2), E = (3,2). We are asked, in that most particular of ways, to join them into one continuous path that uses each dot exactly once and whose consecutive segment lengths are, in order, √10, 5, √10, √10.
First rule of graceful geometry: do not guess. Do the arithmetic. Taste each ingredient before you bake the cake. The distance formula is our recipe: distance between (x1,y1) and (x2,y2) is the square root of (x2 − x1)^2 + (y2 − y1)^2. Often it is tidier to compare squared distances — no radicals hogging the table.
So we make a little list of pairwise squared distances between the five dots. It is like trying each shoe on before committing to the red ones.
AB: (6,3) − (5,0) = (1,3) => 1^2 + 3^2 = 10 => distance = √10 AC: (1,3) − (5,0) = (−4,3) => 16 + 9 = 25 => distance = 5 AD: (6,2) − (5,0) = (1,2) => 1 + 4 = 5 => distance = √5 AE: (3,2) − (5,0) = (−2,2) => 4 + 4 = 8 => distance = √8 BC: (1,3) − (6,3) = (−5,0) => 25 + 0 = 25 => distance = 5 BD: (6,2) − (6,3) = (0,−1) => 0 + 1 = 1 => distance = 1 BE: (3,2) − (6,3) = (−3,−1) => 9 + 1 = 10 => distance = √10 CD: (6,2) − (1,3) = (5,−1) => 25 + 1 = 26 => distance = √26 CE: (3,2) − (1,3) = (2,−1) => 4 + 1 = 5 => distance = √5 DE: (3,2) − (6,2) = (−3,0) => 9 + 0 = 9 => distance = 3
Now breathe and read this like a menu. We have these useful distances among our five dots:
- Distance √10 occurs for AB and BE (two edges).
- Distance 5 occurs for AC and BC (two edges).
- Distance √5 occurs for AD and CE (two edges).
- Distance 1 is BD. Distance 3 is DE. AE is √8. CD is √26.
Here comes the dramatic pause: the requested sequence is √10, 5, √10, √10 — four steps, of which three are √10 steps. To realize that sequence along a path through five points, you need three distinct edges of length √10 connecting successive vertices along the path. But when we look at our distances table, how many √10 edges do we actually have? Two: AB and BE. Two is not three. You cannot pull a third √10 out of a hat because the coordinates are fixed; you cannot invent an extra AB or BE edge without repeating a dot, and our path must use each dot exactly once.
So, with all the charm of a perfectly broken heel on a rainy night, we must admit: the sequence √10, 5, √10, √10 is impossible for these five points. The arithmetic is not being diva — it is simply telling the truth. There are only two √10 edges available, and the requested ordering demands three. That is an elegant and final kind of no.
Before you toss away your stilettos, though, there is a consoling resolution. The same careful checking that rules out this impossible sequence also finds the paths that do exist. For example, if the required sequence was 1, √10, 5, √5 (a different menu), the chain D → B → A → C → E works beautifully:
D (6,2) to B (6,3): squared distance = 1 => distance = 1 B (6,3) to A (5,0): squared distance = 10 => distance = √10 A (5,0) to C (1,3): squared distance = 25 => distance = 5 C (1,3) to E (3,2): squared distance = 5 => distance = √5
It is tidy, it is precise, and it does not ask us to be anything we are not. We used squared distances to avoid messy roots until the final taste test; we used pairwise checks to eliminate impossible orders; and we found a continuous path that uses each dot exactly once.
So what is the lesson, darling? First, the distance formula is not unfriendly — it is your most elegant friend. Second, whenever you see a target sequence of lengths, ask: how many edges of each required length are present among my points? Make that small table first; it is the equivalent of checking your wardrobe before a big night out. Third, use the squared distances (the sums of squares) to avoid unnecessary square roots while you prune possibilities. Finally, if you are stuck, build a little search: pick an edge matching the first length, then from that new vertex look for edges matching the second length to an unused vertex, and so on. Backtrack like the sensible person you are when a choice leads nowhere.
In short — and here is the Carrie piece of wisdom — geometry is a bit like dating: you cannot force chemistry if the numbers do not match. A path demands actual edges that exist, not wishful thinking. When the list of distances asks for more of a particular flavor than your points can supply, you must accept the polite refusal of mathematics and move on to a sequence that does fit, or change the points.
If you want practical hints for practicing Pythagorean Path puzzles:
- Always compute squared distances between every pair of marked points first.
- List which pairs match each required length — these are your only possible edges for that length.
- Count availability: if the target sequence needs three √10 edges and you only have two in your list, the sequence is impossible.
- Use backtracking in a small search: choose an allowed first edge, then allowed second edge to an unused point, and so on. Prune early when remaining edges cannot cover remaining needs.
- Label your points, write coordinates, and present squared-distance checks clearly — neat work is persuasive work.
And because you are 13 and the world is full of possibility: try changing one dot and see how the menu changes. When you alter a coordinate by one or two units, entire sequences that were impossible may become possible. Geometry is merciful that way — sensitive, dramatic, and ultimately responsive.
Now slip on those practical shoes and start measuring. The city awaits, and so do more puzzles. After all, if geometry teaches us anything, it is that the right order of steps makes all the difference.