## Brain-Teaser 453

**From The Sunday Times, 18th January 1970** [link]

Old Hassan has been at it again! He has thought of a new way of parcelling out his square field between his three sons (Rashid and Riad the twins, and Ali).

Calling the three into his tent on the eve of Ramadhan, he addressed them thus:

“My sons, as I am now 74 years of age, I desire that you shall come into part of your inheritance. I have therefore divided my field into four triangular plots such that we each have an area proportional to our age.”

“To avoid any friction regarding upkeep of fences I have also arranged that none of you shall have a common boundary line with either of your brothers. Further, knowing Rashid and Riad’s objection to being too strongly identified with each other, I have arranged for their plots to be of a different shape.”

The twins are 11 years older than Ali.

How old is Ali?

[teaser453]

## Jim Randell 8:40 am

on19 February 2019 Permalink |Let’s suppose that the field is a unit square, and that Ali’s age is

a(a whole number).Then the total of all the ages is

d = a + 2(a + 11) + 74 = 3a + 96, so the corresponding areas of the triangular plots are:Splitting two sides of the square at

x, y ∈ (0, 1)we have:For Rashid and Riad:

and for Ali:

So, given a value for

awe have:So we can consider whole number values for

athat give a solution to these equations:Run:[ @repl.it ]Solution:Ali is 24.A viable solution to the equations is:

so the fields look like this:

Hassan has 37/84 (≈ 44.1%) of the field. The twins have 5/24 (≈ 20.8%) each. And Ali has the remaining 1/7 (≈ 14.3%).

There is a further solution to the equations at:

but this doesn’t give a viable answer.

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