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Sunday Times Teaser 3092 – A Christmas Carol

by Brian Gladman on January 1, 2022

by Nick MacKinnon

Published Sunday December 26 2021 (link)

A Christmas Carol was published on 19/12/1843, when Bob Cratchit was in his forties, almost seven years after Jacob Marley’s death on Christmas Eve. On Marley’s last Christmas Day, a working day for them all as always, Scrooge said to Cratchit and Marley, “We three will work the same dates next year as this year, except that I’ll cover Cratchit’s birthday so he can have the day off, since Marley never works on his.” On Boxing Day, Scrooge decided to allocate extra work dates for next year to anyone whose number of work dates was going to be below the average of the three of them, bringing them up to exactly that average. Daily, up to and including New Year’s Eve, Scrooge repeated this levelling-up to the new average, never needing fractions of a day.

What was Bob Cratchit’s full date of birth?

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  1. Brian Gladman permalink

    The text of this teaser has a flaw since its author wrongly assumed that only one person is subject to the levelling-up process throughout. That assumption does lead to the unique answer but the wording doesn’t support this assumption. This was confirmed by John Owen, who co-ordinates the teasers for the Sunday Times, here.

    This program counts the number of work rosters for which the full levelling-up procedure is both necessary and possible:

    It finds 3429 such work rosters (if ‘all’ is set to false it gives the result for the intended solution).

    Only one of the 3429 possible work rosters acts on only one of Scrooge, Marley and Cratchit:

    which is applied on 26th to 31st December in the year 1836.

    Since they all work on the same dates in 1835 (365 days) and 1836 (366 days) except that Scrooge works on Cratchit’s birthday, this identifies the latter’s birthdate as 29th February. We are also told that Cratchit is in his forties and, unusually, the year 1796 is the only leap year between those in 1792 and 1804. We hence now know that Cratchit’s birthdate is 29th February 1796.

    Despite the teaser’s flaw, in the process of working out the various dates involved it was easy to notice that the wording identifies the unusual eleven year period between the leap years in 1792 and 1804 in which only one leap year (1796) occurs since the normal ‘new century’ leap year is missing. This allowed many solvers, including myself, to guess that the intended solution was 29th February 1796 without even considering the levelling-up process.

    Here is the levelling-up example that I posted here that alerted John Owen to the flawed teaser wording:

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