# Sunday Times Teaser 3062 – Family Stock

*by Howard Williams*

#### Published Sunday May 30 2021 (link)

I have been transferring shares in the family business to my grandchildren, which I’ve done this as part of their birthday presents. On their first birthday I transferred one share, on their second birthday three shares, on their third birthday five shares etc. I have now four grandchildren and at the most recent birthday they were all of different ages. From my spreadsheet I noticed that the number of shares most recently transferred to each grandchild were all exact percentages of the total number of shares transferred to all of them over their lifetimes.

In increasing order, what are the ages of my grandchildren?

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In the light of the discussion below, I have added this analysis.

If the age of the oldest grandchild is \(m\), their percentage \(p_{max}\) of the total number of shares given to the grandchildren must satisfy:\[p_{max}<=\frac{100(2m-1)}{m^2+14}\] Since the percentages for the four grandchildren are all different, this percentage must be at least 4% which leads to the equations \[4m^2+56<=100(2m-1)\] \[(m-25)^2<=586\]which show that \(m<=25\pm\sqrt{586}\). Hence we find that \(m<50\).

@Brian, thanks for showing the” rpartition” function.

How can you be sure there is only one solution?

That really depends on how sure you want me to be. Would I bet £1000 on there

being only one solution? Yes. Would I bet £1,000,000? Not without producing

an analytical solution.

But do I need to be confident that there is only one solution?

Teasers are designed to have only one solution and this means that the only

risk I take in delivering the first solution is that I will have failed to

solve a flawed teaser. And I can live with that! (I did, of course, check

for other solutions).

Brian

It would not be wise to bet someone that they cannot do something.

Either thay can or they are unlikely to honour the bet.

Clearly you do not need to be certain there there is only one solution.

You might or might not want to.

@GeoffR, I think grandchildren are still grandchildren when they become adults.

@Frits: Yes, that is true, but I think they are then probably best referred to as adult grandchildren. My view is that the setter was probably referring to young grandchildren, as evidenced by the ages of the grandchildren in the answer.