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“Suppose there were one nail in each of thefour walls of this room and in addition one nail onthe ceiling and one in the floor,Suppose furtherthat we had to tie strings between these nails.Ihave tWO colors of strings which I use—red andblue.Each connection between any two nails iseither with a red string or with a blue string. “All these strings make up many triangles,that is,any three nails can be considered apexesof a triangle formed by the strings connecting these three nails.My problem is to see if I could distribute the colors so that no triangle has all three sides the same color.”Said young Nicho—las. “This is rather complicated,”mused the mathematician. “It must involve calculations of permutations and combinations,and the like.I didn’t think you knew that much algebra,Nick.”“I don’t,”replied young Nick。“but I can still do the problem.”“Well,all right,tell us how。”said the eld—er. “It is really very simple.”said young Nicho—las,“You only have to know enough to start rea-soning, “The answer is that there will be at least onetriangle all sides of which are the same color.Iwill show that it is impossible to avoid this. “Consider any one nail. Out from it theremust stretch five strings,one to each of the otherfive nails。No matter how you distribute the col—ors in these five strings,at least three of themmust be the same,since you have only two colors.For the sake of argument let assume that three ofthe strings are red. “Consider now the triangle formed betweenthree nails at which the three red strings haveterminated. “If we are to try to avoid a triangle with all three sides the same color,it follows thatthese three nails cannot all be joined to each other with one color.Putting this more simply,thetriangle formed by the three terminal nailsshould not be all blue.At least one of the stringsbetween the three nails must be red.But if so,we have completed a red triangle from the origi—nal nail.”
“Suppose there were one nail in each of the four walls of this room and in addition one nail on the ceiling and one in the floor, Suppose further that we had to tie strings between these nails.Ihave tWO colors of strings which I use—red and blue. Each connection between any any nails iseither with a red string or with a blue string. “All these strings make up many triangles, that is, any three nails can be considered apexesof a triangle formed by the strings connecting these three nails.My problem is To see if I could distribute the colors so that no no triangle has all three sides the same color.” Said young Nicho—las. “This is rather complicated,” mused the mathematician. “It must keep calculations of permutations and combinations, and the like.I didn’t think you knew that much algebra,Nick.”“I don’t,”replied young Nick.“but I can still do the problem.”“Well,all right,tell us how.” “said the eld-er.”It is really very simple.“said young Nicho-las,”You only have to know enough to start rea-soning,“The answer is that there will be at least onetriangle all sides of which are The same color.I will show that it is impossible to avoid this. “Consider any one nail. Out from it theremust stretch five strings, one to each of the otherfive nails. No matter how you distribute the col—ors in these five strings, at least three of the mmust be the same, asnce you have only two colors.For the sake of argument let assume that three ofthe strings are red. “Consider now the triangle formed Wherethree three nails at all the three red strings haveterminated. Formed by the three terminal nailsshould not be all blue.At least one of the stringsbetween the three nails must be red.But if so,we have completed a red triangle from the origi_nal nail.”