How far does the Horse jump in the Chess-Game?
by Klaus-Anton, Jan 21, 2013, 3:36 PM
I have asked a little boy, what he does think, how far the horse jumps in the chess-game. At first he answered me 3. I replied: Two in this direction and one in that direction. Now he estimated: 2 1/4. This is quite good to the correct value.
But it is not so easy to show the right way to find this. When you imagine the chess-horse jumping and begin to count in your imagination, you may think: 3 fields in this direction and 2 fields in that direction. I have made a little LaTeX-picture as attachement, where you can look, to be able to be better concious of the posed problem.
If you think, that the horse jumps three and two fields, it would be as result:
![\[\sqrt{2^2+3^3}=\sqrt{4+9}=\sqrt{13}\]](//latex.artofproblemsolving.com/6/9/7/697059034f7cd896ef391498a50397aa466c14d9.png)
.
But this much too far. Where is the problem?
The horse jumps beginning from the middle of a field and and arrives also on the middle of a field. So you have to compute: 1/2+1+1/2 in the one direction and 1/2+1/2 into the other orthogonal direction, which yields the Domino-Triangle:
![\[\sqrt{2^2+1^2}=\sqrt{4+1}=\sqrt{5}\]](//latex.artofproblemsolving.com/a/9/2/a92a07b9fc0b28fcbcffbc9d6ba1a3633d599ba9.png)
.
How can I give to a child the idea of
using his hands? It now already knows, that it is about 2 1/4. You can tell him a very, very little bit smaller. If you take your third, fourth and fifth finger and these three fingers should show the number 3, you will see: - This does not go correctly. Take your thumb to hold your forefinger in your hand. You now see, that there still is a strange angle of the third finger in relation to the forth and fifth fingers. Imagine this would represent the little small piece, what the horse-jump-length is greater than 2.
But it is not so easy to show the right way to find this. When you imagine the chess-horse jumping and begin to count in your imagination, you may think: 3 fields in this direction and 2 fields in that direction. I have made a little LaTeX-picture as attachement, where you can look, to be able to be better concious of the posed problem.
If you think, that the horse jumps three and two fields, it would be as result:
![\[\sqrt{2^2+3^3}=\sqrt{4+9}=\sqrt{13}\]](http://latex.artofproblemsolving.com/6/9/7/697059034f7cd896ef391498a50397aa466c14d9.png)
.But this much too far. Where is the problem?
The horse jumps beginning from the middle of a field and and arrives also on the middle of a field. So you have to compute: 1/2+1+1/2 in the one direction and 1/2+1/2 into the other orthogonal direction, which yields the Domino-Triangle:
![\[\sqrt{2^2+1^2}=\sqrt{4+1}=\sqrt{5}\]](http://latex.artofproblemsolving.com/a/9/2/a92a07b9fc0b28fcbcffbc9d6ba1a3633d599ba9.png)
.How can I give to a child the idea of
using his hands? It now already knows, that it is about 2 1/4. You can tell him a very, very little bit smaller. If you take your third, fourth and fifth finger and these three fingers should show the number 3, you will see: - This does not go correctly. Take your thumb to hold your forefinger in your hand. You now see, that there still is a strange angle of the third finger in relation to the forth and fifth fingers. Imagine this would represent the little small piece, what the horse-jump-length is greater than 2.Attachments:
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