The only arrangement of the son’s ages that are all different at the father’s death but which would produce consecutive ages if the father’s death had been delayed by a few hours is as follows: \[(a, a+2, a+3, a+4,a+5)\]The ages at the delayed death are then: \[(a+1, a+2, a+3,a+4,a+5)\]The inheritances are proportional to these ages so we can express the Sultan’s fortune \(f\) as: \[f = m(5 a+14) = n(5 a + 15)\] where \(m\) and \(n\) are the inheritance over age ratios for the two sets of ages. We are also told that the one son who would benefit from a delayed death would gain 1000 tesares, which gives:\[n(a + 1) – m a = 1000\] These two equations allow \(m\) and \(f\) to be determined from \(a\) as:\[m=2500\frac{(a+3)}{(2a+7)}\] \[f=2500\frac{(a + 3)(5a+14)}{(2a+7)}\]Since \(m\) is an integer, we are looking for values of \(2a+7\) that are divisors of 2500. Moreover, we know tht \(a\) is at least 17 and, say, less than 120 so the only divisors of 2500 that we need to consider are 50, 100 and 125. Of these only 125 gives an integer \(a\) value (= 59) and hence a Sultan’s fortune of 383160 tesares.

You can investigate this teaser further using Erling Torkildsen’s nice GeoGebra model here.