# Building density *Scale is local* --- A Cintas van pulls up outside a mid-sized factory every week. It drops off clean uniforms, swaps the floor mats, restocks the washroom supplies, refills the first-aid cabinet, and checks the fire extinguishers on the way out. The route gets more valuable with more stops on it, because the fixed cost of the van and driver is spread over more customers, and with more sold at each stop, because getting a van and a person to the door is the expensive bit and once one service has paid for the visit everything else loaded onto it is mostly margin. The second lever travels further than the vans do. The nearest I've run to it was a software business where we cut six products down to two with clear price points and moved the base onto longer contracts; volume barely moved, the value of each sale went up a long way, and the economics of serving a customer changed with it. Put the two together and each feeds the other. Every extra customer and every extra service adds to the density of the base, and the lower unit cost can go back into keener prices or new products in a way a thinner rival can't match. It's part of why Cintas, in a commodity industry, earns an operating margin around 23% while its nearest rival UniFirst earns closer to 8% at present, with the same trucks and the same garments. Some of the gap is scale in plant and purchasing and a richer mix of higher-margin services. The rest is density: Cintas is denser in more markets, so the driving to each stop costs less, and the driving is a large share of a route's day, though the minutes at the door are the same for everyone. Customer density is won road by road. On a given road, whoever has the most customers makes the most stops per mile, so they can charge a fair price and still earn more per stop than a thinner rival, win the next account, and get denser still. Cintas has run that longest and widest, about 12,100 delivery routes out to more than a million customers. You might expect twice the customers on a route to halve the driving between them, and it doesn't work like that. An old result, the travelling salesman problem, says the distance to cover a set of stops grows with the square root of how many there are, so the driving per stop falls with the square root of density. Double the customers and it drops by about a third, and a competitor with half your density drives roughly forty per cent further to every door and can't close that by trying harder. <svg viewBox="0 0 345 158" role="img" aria-label="driving per stop falling with the square root of route density, a rival at half density sits forty per cent higher" style="width:100%;max-width:430px;display:block;margin:0.4em 0 1.1em;"> <line x1="30" y1="130" x2="330" y2="130" stroke="currentColor" stroke-width="0.8" opacity="0.5"/> <line x1="30" y1="130" x2="30" y2="20" stroke="currentColor" stroke-width="0.8" opacity="0.5"/> <path d="M 45 30 C 60 60, 80 82, 105 92 S 170 108, 240 114 S 300 118, 330 119" fill="none" stroke="currentColor" stroke-width="1.6"/> <circle cx="105" cy="92" r="3.5" fill="#c8503a"/> <circle cx="210" cy="111" r="3.5" fill="currentColor"/> <line x1="105" y1="92" x2="105" y2="130" stroke="#c8503a" stroke-width="0.8" stroke-dasharray="3 3" opacity="0.7"/> <line x1="210" y1="111" x2="210" y2="130" stroke="currentColor" stroke-width="0.8" stroke-dasharray="3 3" opacity="0.5"/> <text x="105" y="145" font-size="10" fill="#c8503a" text-anchor="middle">half your density</text> <text x="210" y="145" font-size="10" fill="currentColor" opacity="0.75" text-anchor="middle">you</text> <text x="34" y="18" font-size="10" fill="currentColor" opacity="0.75">driving per stop</text> <text x="330" y="145" font-size="10" fill="currentColor" opacity="0.75" text-anchor="end">customers on the route →</text> <text x="112" y="88" font-size="10" fill="#c8503a">+40%</text> </svg> That is what makes the arithmetic local, and it says something uncomfortable about size. Spread a hundred stops across five equal patches, against a rival who keeps twenty on one, and you are level with them in every patch you share, on five times the customers. Across ten patches you are the one driving forty per cent further. Below, your total stays fixed and you spread it thinner. <iframe src="https://anishshailpatel.github.io/instruments/road-by-road.html" title="Road by road: your stops spread across patches against a rival's twenty on one" width="100%" height="900" style="width:100%;border:0;display:block;margin:1.2em 0;" loading="lazy" sandbox="allow-scripts allow-same-origin"></iframe> > [!note]- The working > The square root is an old and general result: [the shortest route through n stops](https://en.wikipedia.org/wiki/Travelling_salesman_problem) scattered over an area grows with the square root of n times the area, so the driving per stop scales with the square root of area over n. Halve the density and every door costs about 1.4 times more road. The flattening on the right is the quiet part: past a point, more density barely cuts the driving further, so the leader's edge stops growing but never hands anything back. M&A can speed this up, and it's a particular kind of deal. The organic version is slow, because the rival's customers are as sticky as yours, which is why density tends to get bought. Cintas paid roughly $2.2 billion for G&K Services in 2017, whose customers sat on roads Cintas trucks already drove, so laying one network over the other pulled the stops closer together and brought in more than 170,000 customers. In 2026 it agreed terms to do it again, for UniFirst itself. It's the opposite of the bolt-on you buy for its cash flow and leave alone. The whole point is to fold the stops in, which makes it [[Facing into M&A|a change-the-business deal]] from the day it closes, bought at a price that pays the seller for value the buyer has to build. Route overlap is about the closest a synergy comes to earning that, because the saving is arithmetic before it's a plan. Selling more services per stop does something else, too: it makes the customer harder to lose. Cintas holds on to more than nine in ten of its customers each year, and a customer taking uniforms, mats, first aid, washroom supplies and fire protection off one weekly visit has five things to unpick before it can leave. That cuts both ways, and I've been on the wrong side of it: I once took a book of loyal customers as proof the product was fine, and left it a year longer than I should have before admitting it had fallen behind. What holds a customer in place also stops you noticing. The mechanism runs backwards as well. Rentals are billed per uniformed worker, so a downturn that thins headcount thins the base, and the threat to a dense leader is a rival getting denser in one metro, which is another reason the deals get done. Density is the thing to defend, and the map of where you're present doesn't measure it. For a business without vans the road is whatever the fixed cost of serving a customer gets shared across, which might be a vertical that one implementation team knows or a region that one account manager covers. Growth that deepens what you sell to the customers already on the round adds to density, and so does opening a patch you mean to fill, more slowly. Putting a logo in a patch you have no plan to fill dilutes it. Before the next expansion, look at how many of your customers would share a road, because that's the number the square root works on. ---