a        A positive integer is called sumptuous if the sum of any two consecutive digits in the number is either equal to the sum of the first two digits or to the sum of the last two digits. The number 2731546 is not sumptuous, because, although 7+3 = 4+6 and 5+4 = 2+7, 3+1 and 1+5 are not equal to either 4+6 or 2+7.

i      The seven digits from one to seven are rearranged to make a seven digit number (for example, 2731546). Find a way of making the rearrangement so that the seven digit number is sumptuous. Can you find all the possibilities?

ii     Show that, if the six digits from one to six are rearranged to make a six digit number, the number cannot be sumptuous.

 

b       A positive integer is called divisive if the number formed by the first two digits is divisible by two, the number formed by the first three digits is divisible by three . . . and so on. The number 423561 is not divisive, because, although 42 is divisible by two and 423 is divisible by three, 4235 is not divisible by four.

i      The six digits from one to six are rearranged to make a six digit number (for example, 423561). Find a way of making the rearrangement so that the six digit number is divisive. Can you find all the possibilities?

ii     Show that, if the five digits from one to five are rearranged to make a five digit number, the number cannot be divisive.