Christmas gifts
Zoe Mensch
#35 Part 1
By the twelfth day of Christmas, my true love has given me 12 partridges in a pear tree. But which gifts have I received the most of?
Part 2
I want to give all the gifts back. Starting on 26 December 2019, I am going to give one of them to my true love every day. On which date will I give them my final gift?
All squares
David Bodycombe
#36
I met Natalie the other day. She wasn鈥檛 prepared to tell me her age, but she did tell me that in the year N2, she will turn N years old. In what year was she born?
#37
Can you work out (682 鈥� 322)/(592 鈥� 412) without using a calculator? And can you do it without having to square any of the numbers?
Meaningful matches
Paulo Ferro
#38
The figure below has four equilateral triangles. Move two matchsticks to get only three equilateral triangles.
#39
The figure below is composed of 29 matchsticks. Move two matchsticks to get a correct multiplication result.
Puzzle answers
#34 Ant on a tetrahedron
Two spiders patrol up and down a wire each, removing them from the possible routes of the ant. The four remaining wires form a connected path that the final spider can keep going round to eventually catch the slightly slower ant. To prevent the ant quickly darting onto their wires to get behind the chasing spider, the first two spiders should reach the end of their edges to meet the third spider as it goes past.
#35 Christmas gifts part 1
I have received 42 geese-a-laying and the same number of swans-a-swimming. The pattern goes 12 x 1 (partridges), 11 x 2 (turtle doves), 10 x 3 (French hens) and so on, reaching a maximum at 6 x 7 and 7 x 6.
Christmas gifts part 2
23 December 2020. The total number of gifts I received was 1 + (1 + 2) + (1 + 2 + 3) + 鈥� + (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12). Add that lot up and you get 364. If I give away the first gift on 26 December, I give away the last gift 363 days later. In most years, that would be Christmas Eve, but because 2020 is a leap year, it will be a day earlier.
#36 All squares
We are looking for a year that is in the fairly near future and is a square number. The square root of the current year, 2019, is a little under 45, so the next square year will be 452 = 2025, meaning Natalie was born in 2025 鈥� 45 = 1980.
#37
The difference between two squares, a2 鈥� b2, is the same as (a + b)(a 鈥� b).
68 + 32 = 100, and 68 鈥� 32 = 36, so 682 鈥� 322 = 100 x 36.
Similarly, 592 鈥� 412 = 100 x 8.
So (682 鈥� 322)/(592 鈥� 412) = 36/18 = 2.
#38 Meaningful matches
A solution:
#39
