Teaser 3092: A Christmas Carol
From The Sunday Times, 26th December 2021 [link] [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?
In the book The Sunday Times Teaser Book 2 (2023), the “levelling-up” process is described as follows:
… Scrooge decided to allocate extra work dates for the next year to the one whose number of work dates was going to be below the average of the three of them, bringing him up to exactly that average. Daily, up to and including New Year’s Eve, Scrooge repeated this levelling-up for that person to the new average, never needing fractions of a day.
Which is intended to indicate that only one of the three is “levelled-up” each time the process is applied.
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Jim Randell 5:55 pm on 23 December 2021 Permalink |
I’m not sure I understand how this is meant to work as a puzzle.
If Cratchit is in his forties on 1843-12-19, then he was born sometime between 1794 and 1803.
And, if Marley died on 1836-12-24, his last Christmas Day would be 1835-12-25, so Scrooge was planning the dates for 1836.
All we know for sure (for the initial schedule) is that they all work every Christmas Day, Marley never works on his birthday, and Scrooge will work on Cratchit’s birthday in 1836 but Cratchit won’t.
So the number of possible workdays initially scheduled for 1836 is: Scrooge 2-366, Cratchit 1-365, Marley 1-365.
I found 3429 initial assignments of numbers of work days in 1836 that would allow Scrooge to perform his procedure 6 times.
For example if the initial number of allocated work days is (99, 268, 308) we have:
But there doesn’t seem to be any specified conditions to allow us to narrow down these numbers to a set that would allow us to determine a specific date for Cratchit’s birthday.
However, there is a particular date that suggests itself as the only one of the candidate birthdates that has a certain property, so maybe that is the required answer. And it is the same unique date that we get if we make certain assumptions about the behaviour of Scrooge.
Solution: [See below]
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Jim Randell 10:22 am on 1 January 2022 Permalink |
The published puzzle text is flawed.
Apparently the setter intended that only one of the three would be “levelled up” each time the process is applied.
This leads to a single levelling up process:
So, before the “levelling up” process started the work days allocated were 1, 365, 366.
Scrooge must be 366, as he is working an extra day compared to 1835, and there were 365 days in 1835. The extra day worked is the leap day in 1836, and this must also be Cratchit’s birthday.
Cratchit must be 365, as Marley did not work on his birthday in 1835.
Hence, Marley must be the 1. He only worked Christmas day.
So the intended answer was:
Solution: Bob Cratchit was born on 29th February 1796.
And this was the answer I suspected, as it was the only date with a certain property in the required range (i.e. the only 29th February), without worrying about the “levelling up” stuff at all.
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