# Varsity Math, Week 96

## ________________

One team member reads a newspaper article about changes in estate tax, which soon spurs others to create problems about inheritances.

## ________________

### Fair Share

A couple with a mathematical bent has three children and ten grandchildren, to whom their entire estate will be bequeathed. In their will, they leave an equal fraction of their estate to each of the children, and a different equal share to each of the grandchildren. In addition, the inheritance of each grandchild is to the inheritance of each child, as the inheritance of each child is to the entire estate.

What fraction of the estate does a grandchild receive?

### Share and Share A-Different

Another couple has four children and only one grandchild. They also leave all of their estate to these descendants. The shares of the children are the reciprocals of consecutive natural numbers, and the grandchild receives the smallest share.

What fraction of the estate does the grandchild receive?

## Solutions to week 95

Proper Place. Before you get to 2017, there are nine one-digit numbers, 90 two-digit numbers, 900 three-digit numbers, and 1017 four-digit numbers. These take up 9 + 2×90 + 3×900 + 4×1017 = 6,957 digits, and 2017 starts in the next position. So its proper place is 6958.

Earliest Bird. Note that 910 is a early bird; it first appears at position nine, while its proper place is 9 + 2×90 + 3×810 = 2,612, so it is 2,603 places early. Could any number less than 1,000 be earlier than this? Well, the proper place of 899 is 2,586, which is less than the amount 910 is early, so in order to be earlier than 910 a number would have to be in the 900s. What other early birds are there? There’s one every ten: 1920, 2930, and so on. And notice that the early place of each of these is 20 places later, but its proper place is 30 places later, so they are getting earlier and earlier, with 990 being 2,860 – 168 = 2,692 places early. Are there any others? Well, 991 occurs early as 99100, and similarly 992 occurs as 199200. But even with the first of these, 991, its early place occurs 19 positions later than 990 but its proper place is only three positions later, so it is not as early as 990. The situation only gets wors with the 992, 993, and so on. Hence, we conclude that 990 is the earliest bird.

What about that warmup question? Indeed, 2017 is an early bird; it first appears as the last two digits of 1720 followed by the first two digits of 1721. Can you come up with a rule that lets you tell quickly whether or not a number is an early bird?

## Recent Weeks

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

Come back next week for answers and more puzzles.