We found this puzzle in a book published in the 1930s, and so the sums of money involved are much, much smaller than the present day. Nevertheless it is a very interesting puzzle, and we give our thanks to Hubert Phillips.
Please note that at the time of these events there were £5, £10 and £20 notes in circulation. Also at this time pre-decimalisation money was used, with 20 shillings in a pound and 12 pennies in a shilling, with halfpennies and farthings (quarter of a penny) coins in use.
A group of people travelled to London to meet their MP for a tour of the Houses of Parliament followed by lunch. They went by train in a reserved coach with the same number of people in each compartment.
The party and the MP went to a restaurant for lunch, which was three courses for a set price including drinks and tips.
When the MP paid for the meals he handed over one bank note, but embarrassingly was one penny short of covering the bill. Seeing his red face, and knowing him to be a good customer, the proprietor let him off the penny.
How many lunches did the the group plus the MP eat, and (almost) pay for?
Say there were n people having lunch and each lunch cost p pennies, then the total cost must be np pence.
As the MP proffered a single note, but was one penny short, the total bill must have been £5 plus a penny (1201 pennies), £10 plus a penny (2401 pennies) or £20 plus a penny (4801 pennies).
However, 1201 and 4801 are prime numbers which means the bill must have been 2401 pennies.
2401 is 7 x 7 x 7 x 7 so this implies:
343 lunches at 7 pence each
196 lunches at 1 shilling one farthing
98 lunches at 2 shillings and a ha'penny
49 lunches at 4 shillings and a penny
7 lunches at 28 shillings and 7 pence
A party of six occupying an entire reserved carriage is unreasonable, as would be 342 and 195 people in a carriage.
98 people at lunch with 97 on the train just about works numerically but 97 is a prime number and so they could not have split evenly between compartments.
Which means that 49 people had lunch and 48 travelled by train.