so m+n= 137

4 number from 1-9 sum up to 17: 1259 1268 1349 1358 1367 1457 2348 2357 2467 (9 combinations) or 4 number from 1-9 sum to 28: 9874 9865 (2 combinations) The probability= [11* (4!*5!)]/9! =11/126

It would be easier to start with cube, as there is only 5 3-digit cubes (i.e. 125,216,343,512,729). Note that 1 Across and 2 Down is cube. There is two possible solution pair: 125 with 512 or 512 with

The smallest possible sum (a+b)= 36+45=81 1/20= 1/a+1/b= (a+b)/ab --> 20(a+b)=ab b=20a/(a-20)=20+400/(a-20) Hence a+b=a+20-400/(a-20), By differentiation, a+b is minimized when (a-20)^2=400, i.e. a=40,

Yes, they will be free for sure! If number 1 see that 2 and 3's hat is of the same color, number 1 will guess his hat right. If number 1 cannot make a guess, number 2 should know that his hat is different

Answer submitted :)

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My answer is 0.07446 :)