## Math Puzzles (Read the Rules!)

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### Re: Math Puzzles (Read the Rules!)

ChUcK wrote:With only your wits, and a racetrack that can run heats of four horses maximum at once, figure out the four fastest horses.

Question: are we figuring out which four horses are the fastest, and the order of those four, or are we simply finding which four are the fastest without regard to how those four finish?
chiggins
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### Re: Math Puzzles (Read the Rules!)

The four fastest, as well as how they rank. Bukowski needs this info more than he needs his next Red Stripe.

VeganRay, you are incorrect. Charles has already hired a bum.
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### Re: Math Puzzles (Read the Rules!)

veganray wrote:I'll bet you \$100 that if you give me \$200, I will give you \$300 in return.

Would you bet with me? Why?

Ray, read the rules. You can't post another problem until your answer has been confirmed by the asker.
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### Re: Math Puzzles (Read the Rules!)

Sorry, missed that.

After deeper perusal, I have a better solution for 16 horses (A thru P).
• Have 4 random heats. Assume the finishing order for the heats is as follows:
1) A B C D
2) E F G H
3) I J K L
4) M N O P
• Have a fifth heat featuring the winners of the 1st four. Assume finishing order:
5) A E I M
• We now know A is fastest & that either B or E is second-fastest. Have a match race B vs. E. (Heat #6)
• If B wins, the candidates for third-fastest are C & E.
• If E wins, the candidates for third-fastest are B, F, & I.
• Either way, have a race between the possible contenders to decide third-fastest. (Heat #7)
• If B wins, candidates for fourth-fastest are C & E.
• If C wins, candidates for fourth-fastest are D & E.
• If E wins, candidates for fourth-fastest are B, F, & I.
• If I wins, candidates for fourth-fastest are J & M.
• Either way, have a race between the possible contenders to decide fourth-fastest. (Heat #8)

8 heats

I've been trying to formulate a general solution for any values of numbers of horses, max. number of horses per heat, and number (y-fastest) you are trying to determine. Keep running into divide by zero issues. Does anyone know a general solution?
Ryen91 wrote:I am pretty sure I am more intelligent then you think and have allot more knowledge then your post might suggest.

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### Re: Math Puzzles (Read the Rules!)

veganray wrote:Sorry, missed that.

After deeper perusal, I have a better solution for 16 horses (A thru P).
• Have 4 random heats. Assume the finishing order for the heats is as follows:
1) A B C D
2) E F G H
3) I J K L
4) M N O P
• Have a fifth heat featuring the winners of the 1st four. Assume finishing order:
5) A E I M
• We now know A is fastest & that either B or E is second-fastest. Have a match race B vs. E. (Heat #6)
• If B wins, the candidates for third-fastest are C & E.
• If E wins, the candidates for third-fastest are B, F, & I.
• Either way, have a race between the possible contenders to decide third-fastest. (Heat #7)
• If B wins, candidates for fourth-fastest are C & E.
• If C wins, candidates for fourth-fastest are D & E.
• If E wins, candidates for fourth-fastest are B, F, & I.
• If I wins, candidates for fourth-fastest are J & M.
• Either way, have a race between the possible contenders to decide fourth-fastest. (Heat #8)

8 heats

I've been trying to formulate a general solution for any values of numbers of horses, max. number of horses per heat, and number (y-fastest) you are trying to determine. Keep running into divide by zero issues. Does anyone know a general solution?

Why are you only racing two horses in a heat?
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### Re: Math Puzzles (Read the Rules!)

veganray wrote:I'll bet you \$100 that if you give me \$200, I will give you \$300 in return.

Would you bet with me? Why?

Of course not, because if you win the bet I break even, and if you lose the bet you get \$100 :D

I'm not posting another problem because this was a rogue submission.
We're at our best when it's from our hips
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### Re: Math Puzzles (Read the Rules!)

Parks wrote:Why are you only racing two horses in a heat?

I didn't see a prohibition of races containing <4 horses in the problem, and 2 (or 3) is enough to solve the problem using my "tree" algorithm.
Ryen91 wrote:I am pretty sure I am more intelligent then you think and have allot more knowledge then your post might suggest.

Cheers & chings!
Vegan Ray
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### Re: Math Puzzles (Read the Rules!)

veganray wrote:• We now know A is fastest & that either B or E is second-fastest. Have a match race B vs. E. (Heat #6)

I'm not sure that's right, for example:

Code: Select all
`16 15 14 13 9 10 11 12 8  7  6  5 4  3  2  1`

becomes:

Code: Select all
`13 14 15 16 9 10 11 12 5  6  7  8 1  2  3  4`

becomes:

Code: Select all
` 1 14 15 16 5 10 11 12 9  6  7  813 2  3  4`

So after your first five heats, you cannot say that B or E are the second fastest, right?
chiggins
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### Re: Math Puzzles (Read the Rules!)

After heat #1, we know A is faster than B, C, & D
After heat #5, we know that A is faster than E, I, & M
E is faster than everybody else in heat #2 (he won)
I is faster than everybody else in heat #3 (he won)
M is faster than everybody else in heat #4 (he won)
From the results of heat #5, we know that E is faster than I or M
Therefore we know that E is faster than everybody in heats 2, 3, & 4
From the results of heat #1, we know that B is faster than C or D
Therefore, A is fastest, & either B or E must be second-fastest

BTW - if the results of heat #5 were different, we would just rename the first four heats such that heat #5's winner's heat is deemed "heat #1", heat #5's runner-up's heat is deemed "heat #2", heat #5's 3rd place horse's heat is deemed "heat #3", and heat #5's loser's heat is deemed "heat #4".
Ryen91 wrote:I am pretty sure I am more intelligent then you think and have allot more knowledge then your post might suggest.

Cheers & chings!
Vegan Ray
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### Re: Math Puzzles (Read the Rules!)

(To be clear, my matrices are just one possible result, working backwards from knowing the 1st to 16th fastest horse.)

veganray wrote:After heat #1, we know A is faster than B, C, & D

That's reflected as the 13th fastest horse beating 14, 15, and 16th fastest. Okay.

E is faster than everybody else in heat #2 (he won)

That's reflected as the 9th fastest horse beating 10, 11, and 12th fastest. Okay.

I is faster than everybody else in heat #3 (he won)

That's reflected as the 5th fastest horse beating 6, 7, and 8th fastest. Okay.

M is faster than everybody else in heat #4 (he won)

That's reflected as the 1st fastest horse beating 2, 3, and 4th fastest. Okay.

After heat #5, we know that A is faster than E, I, & M

I moved this down into chronological position, and it's reflected as a race between the 1st, 5th, 9th, and 13th fastest horses, with the first horse coming in 1st.

From the results of heat #5, we know that E is faster than I or M

As far as I can tell, this only means that the 5th fastest horse is faster than the 9th or 13th fastest horse.

Therefore we know that E is faster than everybody in heats 2, 3, & 4

That's the trouble spot, it may be that I'm dim and not seeing it, but looking at my example all I can say for sure is that you can be %100 sure that the fastest horse on the track won heat #5. The second, third, and fourth horses were beaten in heat #4 by the fastest horse, but I don't see how they've been accounted for in heat #5?

BTW - if the results of heat #5 were different, we would just rename the first four heats such that heat #5's winner's heat is deemed "heat #1", heat #5's runner-up's heat is deemed "heat #2", heat #5's 3rd place horse's heat is deemed "heat #3", and heat #5's loser's heat is deemed "heat #4".

Heh. I do not understand "we would just rename the first four heats such that heat #5's winner's heat is deemed "heat #1"", apologies for my obtuseness.
chiggins
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### Re: Math Puzzles (Read the Rules!)

Veganray's solution is correct. He presented it in a way I have never seen before, but the logic works.

edit: Here's my solution, which is the same as veganray's but it keeps four horses in each heat:

race 4 heats of all different horses. here are theoretical results (in order of placing)
heat 1: ABCD
heat 2: EFGH
heat 3: IJKL
heat 4: MNOP

Then race the winners from each heat with the theoretical results
heat 5: AEIM

That makes A the fastest and it eliminates horses H,L,P

heat 6: BEIM and the winner is the second fastest (assume B wins) and this eliminates horses G,K,O.

heat 7: EIM and the winner is the 3rd fastest (assume C wins) and this eliminates horses F,J,N

heat 8: DEIM and the winner is the 4th fastest. Assume D wins, which gives A>B>C>D>all other horses.

8 races is the minimum.

My solution makes some horses run unnecessarily. Certainly heat 6 only needs to be between B and E, as they are the only possible candidates for 2nd fastest.
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ChUcK
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### Re: Math Puzzles (Read the Rules!)

Well, Bukoswki might've just hired the bum instead of you, because using a different method it only requires a 7th heat a fair percentage of the time :D

Unrelated side note: keep in mind that HKLNOP were all eliminated after the 5th heat as well, not just HLP.
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### Re: Math Puzzles (Read the Rules!)

veganray wrote:BTW - if the results of heat #5 were different, we would just rename the first four heats such that heat #5's winner's heat is deemed "heat #1", heat #5's runner-up's heat is deemed "heat #2", heat #5's 3rd place horse's heat is deemed "heat #3", and heat #5's loser's heat is deemed "heat #4".

Now I got it. I was working columns without bringing the other finishers with them. Once I turned them into chains with greater-thans between the horses, and moved the whole chain up or down the stack, it made perfect sense. (Was like a hand came down and POW I got illuminated.) Helluva puzzle and nice work, doctor.
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### Re: Math Puzzles (Read the Rules!)

Parks wrote:Unrelated side note: keep in mind that HKLNOP were all eliminated after the 5th heat as well, not just HLP.

Whoops, I think you're right. I think that after the 5th race HKLNOP are all at least 4 "greater than"s from A, therefore eliminated from competition.
We are not like those other golfers. We throw our clubs and keep our balls where they belong. -Ol' Bob
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### Re: Math Puzzles (Read the Rules!)

Does that mean I'm supposed to post the next brainteaser?
Ryen91 wrote:I am pretty sure I am more intelligent then you think and have allot more knowledge then your post might suggest.

Cheers & chings!
Vegan Ray
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