The Mathematics of Mushing

Sled dogs are hitched up in pairs along the gangline, from the leaders to the swing dogs to the team dogs to the wheelers. Many of the dogs can run in any position, including lead, and mushers will rotate their dogs among the positions to avoid repetitive motion injuries, boredom, and fatigue.

However, for every unique race circumstance, every dog team has a "one best arrangement" of dogs to insure the optimum performance level of each animal and of the team as a synergistic whole.

The successful musher is the one who knows what that one best combination of dogs is.  The astonishing thing is just how many choices mushers have to select from.  It boggles the mind!

If you are skijoring behind one pet dog, you only have one choice because you only have one dog to fill one position.  If you skijor behind two dogs, you can put one dog in either of two positions, on the right or on the left.  The other dog is stuck with the remaining position.  Once you put King on the left, Queenie has to go on the right.  Or, if King runs on the right, Queenie has to run on the left.  You have two choices for King with one remaining spot for Queenie.  Two times one equals two (2 x 1 = 2).  See, math is simple!

Now, if you are running a little, four-dog team, you have four places to put King:  Left lead, right lead, left wheel, or right wheel.  Once you choose King’s position, there are only three spots left for Queenie:  If King is in left wheel, Queenie can go beside him in right wheel, or ahead of him in left lead, or across in right lead.  If King starts out in right lead, then Queenie must go either in left lead, right wheel, or left wheel, etc.

In other words, there are four places to hook King and three remaining places for Queenie. This means there are four times three, or twelve (4 x 3 = 12), different ways you can hook King and Queenie into a four-dog team.  But, there are still two open places.

For each of the 12 different pairings of King and Queenie, there are two spots left in the team for Prince.  If King is in right lead and Queenie is in left lead, Prince can only go in right or left wheel.  If King is in left wheel and Queenie is in right lead, Prince can go in either left lead or right wheel, etc.  Get the picture?  Thus, there are twelve times two, (or four times three times two), or twenty-four, different combinations of King, Queenie, and Prince (4 x 3 x 2 = 24) in a 4-dog team.

For each of the twenty-four combinations of these three dogs there is only one spot left for poor Duke.  A musher running a little 4-dog team, has a whopping twenty-four (4 x 3 x 2 x 1 = 24) possible combinations to choose from for that one best alignment of King, Queenie, Prince, and Duke.  Wow!  And, that assumes the conditions don’t change during a long race, like open water, or steep hills, or road crossings, or scary spectators.  Such changes might require yet a different alignment of dogs.  More wow!

This method of combining objects in every possible combination, without repeating any objects twice, is called a "factorial" in mathematics, and it is written as "n!"  Writing "n!" is the same as saying "4 x 3 x 2 x 1 = 24."  And, you can start with any number for "n."

Now, if the musher graduates to a six-dog team, the number of possible combinations of those six dogs for six team spots is 6! (six factorial), or 6 x 5 x 4 x 3 x 2 x 1 = 720. There are seven hundred and twenty different ways of arranging six dogs in a six-dog team.

When the Yukon Quest 300 begins, each team will leave the starting line with at least a minimum of eight dogs.  Each musher has to determine where each dog will run at its best.  The dog driver has to choose from among at least eight factorial possibilities (8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320). That’s forty-thousand, three hundred and twenty choices based on the musher’s experience and dog savvy.

The Quest Marathon mushers will be faced with the same decision before they leave. These mushers will lean back against their trucks to ruminate over the many possible ways in which they can arrange their 14-dog teams:  Who should be the leaders, who only runs on the right side, who fights with whom, who is afraid of crowds, who is not afraid of overflow on the river, who passes other teams easily, who drives up steep grades, who might be coming into heat, and on and on and on...

Each team will leave with 14 dogs, with each dog running in its optimum position, and that alignment will be chosen by the musher from among a possible 14! choices of alignments.  And, fourteen factorial equals 14 x 13 x 12 x……x 3 x 2 x 1 = 87,178,291,200. That’s eighty-seven billion, one-hundred and seventy-eight million, two-hundred and ninety-one thousand, two hundred different ways of hooking up that 14-dog team.

You’d better know your dogs, if you’re going to be a successful musher.

There are other races allowing dog teams larger than the 14-dog limit of the Yukon Quest.  The Iditarod allows mushers to start with 16 dogs.  The major sprint races in the far North, like the Rondy and the North American in Alaska, include an Unlimited class, where any number of dogs can be run in one team.  In a sprint race covering only 12 to 30 blistering miles, a mistake of only one or two seconds can cost a musher the race.  There’s no time to stop and rearrange a dog team in a sprint race if you’ve hooked up the wrong dog in the wrong position.  Therefore, the sprint musher has only one chance to assemble the best combination of dogs and that is before the start.

When a musher assembles his best combination of 16 dogs, he does so from a possible pool of 16 factorial ways of hooking up his crew. That’s over 20 trillion (16! = 20,922,789,890,000) different combinations, and a wrong choice could cost him the winner’s circle.

A World Champion sprint musher from Canada once hooked up 28 dogs for an Unlimited sprint race.  Now, the more dogs in a team running 20 miles per hour, the greater the chances that the wrong dog in the wrong position will slow the team and lose the race.  That musher had to know his dogs well enough to organize the best alignment of dogs from among 28 factorial choices. And 28 factorial equals over three hundred billion, billion, billion (28! = 304,888,344,600,000,000,000,000,000,000) different possibilities!

Of course, he IS a Canadian.

Dr. Jerry Vanek has been a musher or sled dog race veterinarian for the past 30 years, including five Yukon Quests.  He is a former officer of the ISDVMA and he continues to write and speak widely on the subject of sled dog medicine.

Jerry Vanek, DVM