Sunday Times Teaser 2783 – Old Boys’ Banquet

by Danny Roth

George and Martha have arranged the seating plan for the annual Old Boys banquet; it involves a number of tables, each seating 50 diners. More than half the Old Boys are bringing one female guest each and the rest are coming alone. Martha wrote down three numbers, namely the number of Old Boys bringing a guest, the number of Old Boys coming alone, and the total number of Old Boys coming. George noted that the three numbers between them used each of the digits 0 to 9 exactly once.

How many Old Boys are bringing a guest, and how many are coming alone?

1. A bit of analysis provides a much faster solution:

2. Using the code below I seems to get two valid answers

3. 4. 5. I took a short cut using Brian’s letter/number substitution Python program on his ‘paper and pencil’ teaser site. If $$efg$$, $$hij$$ and $$abcd$$ are the number of boys with and without partners and the total respectively, then: $$efg + hij = abcd$$ where $$efg > hij$$. The total number of diners is $$2efg + hij$$ which equals $$50N$$ where $$N$$ is the number of tables and $$2g+j=10$$. This gave 2 possible solutions of which only $$752 + 346 = 1098$$ satisfied (i). (1950 guests, 39 tables.) Putting 2g + j = 20 gave no solution. Similarly putting $$efgh$$ and $$ij$$ as the number of boys and $$2h + j$$ equal to 10 or 20 quickly gave the other 3 solutions.