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Sunday Times Teaser 3062 – Family Stock

by Brian Gladman on May 28, 2021

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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7 Comments Leave one →
  1. Brian Gladman permalink

    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 \(m<50\).

    • Frits permalink

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

      How can you be sure there is only one solution?

      • Brian Gladman permalink

        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).

        • Tony Smith permalink

          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.

  2. GeoffR permalink

    • Frits permalink

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

  3. GeoffR permalink

    @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.

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