#251 Children of Quirkopia
(set on 9 December)
Solution
The answer is (b) the population is still divided 75 per cent male to 25 per cent female. This is true regardless of the make-up of individual families. To see why, imagine visiting the maternity hospital. There is a 75 per cent chance that the next baby to be born will be male. If so, then that family isn鈥檛 allowed more children, but this is irrelevant to the midwife, who is awaiting the arrival of the next baby, which is also 75 per cent likely to be a boy. Each new baby鈥檚 sex is independent of what happened previously, so the draconian policy has had no impact on the ratio of males to females; it merely reduced the birth rate.
#252 Santa clocks
set by Brian Hobbs
'Twas the night before Christmas, when all through the house,
Not a child was sleeping; just me and my spouse.
For Santa was coming, and kids needed proof
So they listened intently for hoofs on the roof.
They had all been asleep when we turned out the light,
But each woke up once in the course of the night.
First Jill, and then Jennifer, Jimmy and Jake
Spent no more than 6 hours lying awake.
The only sounds heard were the gentle tick-tock
And the hourly chimes of the grandfather clock.
At the top of the hour, the clock would make note
By chiming the number the hour hand showed.
As each child listened, they passed the time
By totalling up the grandfather clock chimes.
They tallied them to a particular number
'Til finally succumbing to sleeping and slumber.
None of the kids, in regards to the timing
Fell asleep or awoke while the hour was chiming.
Through different hours did each child count
Yet they each wound up totalling the same amount.
"What are the odds?" we all asked one another.
This number they counted is yours to discover
Are there four sets of consecutive hours
That make the same sum? Use your deductive powers!
"Next year," the kids thought, "we won't even try it;
Santa Claus seems to be terribly quiet."
By evening the kids had all leapt out of sight
"Merry Christmas, but we're off to bed, so good night!"
#252 Santa clocks
Solution
They each counted 15 chimes, broken up like this: (12, 1, 2), (1, 2, 3, 4, 5), (4, 5, 6) and (7,8). Every odd number can be summed with two consecutive integers (such as 7, 8). Every number divisible by 3 can be summed with three consecutive integers (such as 4, 5, 6), and every number divisible by 5 can be summed with five consecutive integers (such as 1, 2, 3, 4, 5). The number 15 is an odd number that is a multiple of 3 and 5, and can also be made by summing 12, 1 and 2. That makes 15 the smallest number that can be summed in four ways with consecutive numbers on a clock. The next such number is 33, but that requires one child to count for over 7 hours, contradicting the "no more than 6 hours" that any one child was awake.
#253 Two tenors
set by Daniel Griller
Starting at the number 1, as you embark on a festive stroll along the number line, you will, from time to time, land on whole numbers whose digits sum to a multiple of 10; like 46 or 785, or 90939. Let's call these special numbers "tenors".
Sometimes, the gap from one tenor to the next is small, like the time it takes for Santa's sleigh to fly across the sky, while at other times you have to be patient, much like how children must wait on Christmas Eve. But rest assured, no matter how far you have walked, there is always another tenor waiting for you to arrive, just like the next surprise gift under the tree.
The question is: what is the largest possible gap between one tenor and the next, and how soon into your walk will you encounter such a gap?
#253 Two tenors
Solution
The smallest tenor is 19. After this, in each block of ten numbers (20-29, 30-39 and so on), the digit sum increases one by one, so exactly one number in each block has a digit sum divisible by 10. That is, each block contains exactly one tenor. Given this, the biggest gap we could hope for is 19 鈥� when the first number in one block is a tenor and the last number in the next block is also a tenor. It turns out this is possible, and the first appearance of such a gap is between 280 and 299.
#254 Elemental
set by Paul Board
To crack this puzzle, each cell may have one or two letters in it, and each of these must constitute an elemental symbol. Each clue corresponds to the relevant line of the tree. Once you have solved the clues and completed the tree (including the baubles), rearrange the letters in those baubles (without switching the letters in a two-letter elemental symbol) to form something you may use in your Christmas cooking or in a potpourri (two words: 8,5). You might be advised to have a copy of the periodic table to hand!
1 Said three times when Santa Claus laughs (2)
2 __ Man, a character in The Wizard of Oz (3)
3 Winter precipitation and the surname of a chemist and novelist famous for his lecture on "The Two Cultures" (4)
4 Nuts that form the basis of marzipan in a Christmas cake (7)
5 German chemist who was born on Christmas Day 1876 and awarded the Nobel prize for chemistry in 1928 (7)
6 Alcoholic component of Buck's Fizz (9)
7 The __, a ballet by Pyotr Ilyich Tchaikovsky, often performed at Christmas (10)
8 One of the Christmas gifts from the three wise men (12)
Answers to these puzzles, plus the solutions to 9 December's puzzle and crossword, on page 78
#254 Elemental
Solution
The final answer is:(C)(In)(N)(Am)(O)(N)(S)(Ti)(C)(K).