Jack Handley's new telephone number, together with the area code, forms a ten-digit number consisting of a pair of 1's separated by one digit, a pair of 2's separated by two digits, a pair of 3's and a pair of 4's both separated by three digits, and a pair of 5's separated by five digits. Neither the sixth nor the ninth digit is a 4.

 

What is his new telephone number?

Jack Handley's new telephone number' date=' together with the area code, forms a ten-digit number consisting of a pair of 1's separated by one digit, a pair of 2's separated by two digits, a pair of 3's and a pair of 4's both separated by three digits, and a pair of 5's separated by five digits. Neither the sixth nor the ninth digit is a 4.

 

What is his new telephone number?[/quote']

 

It's 4352432151. Got there mostly by chance, the only calculation I did was assign x,y,z,w and k as the positions of the first 1,2,3,4 and 5 respectively. Then it's clear that the positions of the second 1,2,3,4 and 5 are x+2,y+3,z+4,w+4 and k+6 respectively. The sum of all the positions is 1+2+...+10=55, so we get that x+y+z+w+k=18. The first five positions that add to 18 are 1,2,3,4 and 8. I didn't know how to go on more formally from here so I tried a couple of combinations to arrive at the solution.