# Which horse wins the race?

Freddy, Charlie, Apple, and Boris are each one of four horses in the Belmont Stakes, a 1 1⁄2-mile race. After a mile the horses are in the following positions:

• Freddy is at least an eighth of a mile ahead of Charlie, Apple, and Boris.
• Boris is behind Apple, but by less than an eighth of a mile.
• Charlie and Apple are tied.

For the rest of the race the horses will move accordingly:

• Boris will move at 30 mph.
• Freddy will move at 25 mph.
• Charlie and Apple will move at 27mph each.

Which horse wins the race?

• Cheat answer: if Freddy is one mile ahead of everyone else then he easily wins, assuming this puzzle has a single correct answer then he must always win.
– ffao
Sep 10 '16 at 16:33
• Can we not suppose that Freddy is 7/8 of a mile ahead of the others? It's possible by these criteria that Boris never left the starting line and Apple/Charlie are both only 1/8 of a mile ahead of him when Freddy reaches the one mile mark. Edit: @ffao had the same idea as me. Sep 10 '16 at 16:34
• Also a cheat answer: If there is a calculable scenario to this puzzle where Freddy does not win, then the only other horse that can win is Boris because Charlie and Apple have moved at the same pace the entire race and will tie with each other at the finish line. Sep 10 '16 at 16:38
• I think there's enough of a 'trick' / interesting approach to this that it's a maths puzzle and not just an exercise. Voting to leave open. Sep 10 '16 at 18:54
• Is "after a mile" a well known term from horse racing? Because I don't understand what it means. Does it mean "After at least one horse has run a mile"? Does it mean "After every horse has run at least a mile"? Sep 10 '16 at 22:03

• Boris moves 3 mph faster than Apple and Charlie, and the distance between them is less than $\tfrac{1}{8}$ miles, so he will overtake them in less than $\tfrac{1}{24}$ hours.

• Boris moves 5 mph faster than Freddy, and the distance between them is at least $\tfrac{1}{8}$ miles, so Boris will take at least $\tfrac{1}{40}$ hours to overtake Freddy.

• Freddy moves 2 mph slower than Apple and Charlie, and the distance between them is more than $\tfrac{1}{8}$ miles, so they will take at least $\tfrac{1}{16}$ hours to overtake him.

• Both $\tfrac{1}{40}$ and $\tfrac{1}{16}$ are greater than the times any of the horses will take to run half a mile ($\tfrac{1}{60}$, $\tfrac{1}{50}$, and $\tfrac{1}{54}$ for Boris, Freddy, and A&C respectively), so none of the horses will overtake each other before the finish line.

• Beat me just bearly Sep 10 '16 at 16:49

First note that Apple and Charlie do the exact same things during the contest.

Let's calculate how much time each of the horses need to finish the game. From physics we know that t=s/v, that is the time taken to reach a distance is proportional to velocity. With this formula we can deduce that, with the obvious notation of subscripts,

t_F=s/v=0.5/25=1/50=0.02

t_A=0.5/27+d_A/27>=5/216=0.023>0.02=t_F

t_B=0.5/30+d_A/30+d_B/30>1/48+d_B/30>0.02=t_F

Thus, irrespectively is of how far Boris is behind the others, Freddy wins.