On Friday I posted this puzzle….

I have a cage in my garden and it contains only chickens and rabbits (I don’t really, but play along). There are 72 heads and 200 feet in the cage. How many chickens are there, and how many rabbits?

If you have not tried to solve it, have a go now.  For everyone else the answer is after the break.

There are 44 chickens and 28 rabbits in my cage.  Did you solve it? And if so, how?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. Eddie says:

I solved it by inverting a 2 by 2 matrix.

1. Lazy T says:

? xirtam 2 yb 2 a

2. du Lisoy says:

X+Y = 72
2X+4Y = 200
Y being the number rabbits…

1. ChrisR says:

and for those who want it spelled out (who can’t remember ‘simultaneous equations’)..
X+Y=72 means 2X+2Y = 144
so the difference between this and du Lisoy’s last equation is
2Y on the left and 56 on the right, so there are 28 rabbits.

X + 28 = 72 means X (chickens) = 44

Given the numbers involved (and knowing you wouldn’t get an answer including 1/2 a rabbit) it was probably easy enough to do in one’s head, but there’s a choice between doing it on paper straight-away and knowing you’re on the right track, and the risk of starting off doing it in your head but having to resort to pen/paper if the numbers got awkward.

I’m interested in seeing what short-cuts there may be (and not trial and error).

2. (and knowing you wouldn’t get an answer including 1/2 a rabbit)

Mine did because a blasted fox got in over night!

3. Andrew says:

If all 72 animals had two legs there would be 144 feet, so the other 56 feet must belong to the rabbits. Two extra feet per rabbit, means 28 rabbits.

1. Stevie says:

That’s a brilliant, simple and logical reasoning! I’m afraid that I got the answer only after much trial and error with the grey cells, pencil and paper and eventually excel.

2. Lazy T says:

That’s how I got it, but I wouldn’t be able to explain it as succinctly as Andrew did.

3. @Sceptic_Tank says:

That’s how I did it too.

4. My solution too, except that I reduced it to 100 pairs of feet at the start so I didn’t have to do the hard multiplication and division bits 🙂

5. ChrisR says:

hats off to you

6. Peter says:

That is a lovely solution…I went the dogged R+C=72, 4R+2C=200 route…yours is far more elegant.

4. VicinCA says:

x = # of rabbits (4 legs)
y= # of chickens (2 legs)

Use simultaneous equations to solve for the # of rabbits.
x + y = 72
4x + 2y =200

4x + 2y = 200
2x + 2y = 144

Subtract the equations.
2x = 56
x = 28 rabbits

72 – 28 = 44 chickens

5. 72 heads and 100 pair of feet. That’s 28 pairs too many in case of only chickens, which makes 28 rabbits. 72 – 28 = 44 chickens. Did not need an (advanced) pen or paper.

6. The number of left feet is 100 and equals the number of chickens plus twice the number of rabbits.
The number of heads is 72 and equals the number of chickens plus the number of rabbits.
Thus the difference of 100 – 72 is the number of rabbits.

I was able to do that without inverting a matrix or writing down two equations with two unknowns. (Actually I’ll have to admit the latter is how I did it on Friday).

7. zippy says:

72 heads is 72 animals. If they were all chickens that would be 144 feet, which is 56 too few. Therefore 28 of the chickens need to be replaced with rabbits, leaving 44 chickens.

Interesting to see the split between people who thought it through simply like this and those who saw it in the form of pen and paper equations.

8. Roland says:

I had 16 chicken and 56 3-legged rabbits (their fourth leg was sold as lucky rabbit’s foot, the poor animals…).

1. ivan says:

I accidentally solved that problem first too. Then I remembered rabbits have 2 more legs than chickens, and halved the 56 (=200-144). That’s why it took me 10 seconds instead of 3 seconds.

I’m disappointed that Richard didn’t give the simple explanation why the answer is rabits = 100-72. Looks like he solved simultaneous equiations too.

9. I found the same answer, but instead of using equations like the rest of you, I used an estimate. I asked: what if 100 of those feet belonged to chickens, and the other 100 to rabbits, how many heads would I have? The answer turns out is 75, which is only 3 off. Plussing and minussing lead me to the correct answer within seconds.

10. pete says:

1. Hugh Janus says:

Pete, it’s already bad enough mixing up legs, but mixing up toes take this to another level

2. Slange Navarr says:

Another world-beater, Janus. A comedy masterclass with every post. Thank you, as ever.

3. Hugh Janus says:

Anagram of Slange Navarr = A graven snarl

4. Slange Navarr says:

Side-splitting, Janus. The quality is getting better and better, even after I thought you’d peaked with that hilarious pun on Friday.

5. Hugh Janus says:

Slainte mhath!

6. Slange Navarr says:

Yes, obviously, funnyman. But keep going with the anagrams. Not only are they a comedy gift to mankind, you might actually learn something. Call it my return gift to you.

11. lindamp says:

Assuming each chicken has 1 head and 2 feet, and each rabbit has 1 head and 4 feet:

Each animal has at least 2 feet. 72 animals.
72 * 2 = 144 feet

200-144 = 56 “extra” feet

56 extra feet = 28 rabbits with 4 feet.

Therefore 28 rabbits and (72-28 =) 44 chickens.

1. Nat says:

ha-ha- you assumed chickens and me rabbits as chicken it is a bird (i was taught like this in old school 20 years ago) and not the animal, so 72 x 4 = 288
288-200=88
88:2 (legs of chickens)= 44
so 72 – 44-= 28
well it took me some time

12. Julia says:

This puzzle was a revelation to me. I was on the phone (listening to the various options for buttons to press) when I read and decided to use algebra (same method as that used by du Lisoy and VicinCA) as I wouldn’t be able to concentrate hard enough for trial and error. If anyone had asked me what’s easier – algebra or trial and error – I always would have opted for trial and error. Turns out I was wrong.

1. Julia says:

13. Duncan says:

Some very elegant solutions, but apart from using matrices aren’t they all just simultaneous equations?

14. issahamati says:

wow

15. L.Long says:

But he said….
‘There are 72 heads and 200 feet in the cage.’
So obviously they had a great BBQ and there are no chickens or rabbits left in the cage. So the answer is 0.

16. I did it through the “extra” legs method…. but then did it by simultaneous equation, except I solved the equations a slightly different way than the others shown

c + r = 72
2c + 4r = 200 which I auto simplify almost instantly by dividing by 2…

c + 2r = 100

isolate c…

c = 100 – 2r

plug it into the first as the value of c

100 – 2r + r = 72

100 – r = 72

-r = 72 – 100

-r = -28

negatives cancel

r = 28;

1. ChrisR says:

isolating ‘r’ from the first two equations, by taking the first from the auto-simplified second, is a little simpler and is actually how the ‘elegant solvers’ did it in their heads – even if it wasn’t called simultaneous equations

17. I used simultaneous equations…..at least 30years since I last did them!!! But totally with keithnicholas is using r & c instead of x & y. Teaching my son at the moment and using algebra that has more meaning….r=rabbit & c=chicken just brings maths and puzzles more to life and into the real world……or is that just me?

18. Bill says:

I multiplied the total num of heads with the feet of the rabbits and it was a remainder of 88 feet. So, I thought that 88 would be the feet of the chickens and therefore 44 heads.

19. I used two equations
72=C+R
200=2C+4R or 100=(C+R)+R

Then plugged the first into the second for 100=(72)+R, which gives 28 rabbits.

1. Very fun puzzle, btw. I love a reason to use algebra.

20. Like nearly everyone else I just used algebra 2C + 4(72-C) = 200 where C= number of chickens.

21. Anonymous says:

x+y=72
4x+2y=200
Isolate y by subtractracting x.
y=72-x
4x+2(72-x)=200
4x+144-2x=200
Combine like terms
4x-2x=2x
2x+144=200
subtract 144
2x=56
divide by 2
x=28
Then plug back into original equation.
x+y=72
28+y=72
subtract 28
y=44
x=rabbits=28
y=chickens=44

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