First I’d like to say THANK YOU to the hundreds of people who left solutions at the bottom of the question post, on the Guardian Facebook page, on Twitter with the hashtag #MondayPuzzle and in emails to me.
I’ve been totally overwhelmed by the quantity and variety of solutions - and quite embarrassed that my own was so boring!
Evidently there is no unique solution - I was half hoping a computer scientist would let me know exactly how many solutions there are with only the four basic operations. Maybe someone will...
Before we get to my favourites, I’ll explain how I did it myself. Since a fair amount of you were struggling.
The way to solve this type of puzzle is through “enlightened” trial and error. My approach always begins by factorising the year, in this case 2016. Factorising means dividing in to smaller and smaller pieces so that all is left is a string of prime numbers.
2016 breaks down into 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7, which is the same as 25 x 32 x 7
When I saw this I thought that I would leave 7 as it is, and then try to make 25and 32 (32 and 9) out of (10 9 8) and (6 5 4 3 2 1).
If this worked I’d have (10 9 8) x 7 x (6 5 4 3 2 1) = 2016
I quickly noticed that 10 – 9 + 8 = 9. And then playing around that
(6x5) - 4 + 3 + 2 + 1 = 32.
So I had a solution: (10 – 9 + 8) x 7 x ((6x5) – 4 + 3 + 2 + 1) = 2016
It is a pretty dull solution though. The most elegant has to be this one, tweeted earlier by James Annan (and subsequently by others):
1) √(x+3) = 2 + √(7-x) Область определения: -3<=x<=7 Возводим всё в квадрат x+3=4+2√(7-x)+7-x Оставляем корень отдельно. 2x-8=2√(7-x) Делим всё на 2 √(7-x)=x-4 Поскольку корень арифметический, то область определения 4 Опять возводим в квадрат 7-x=(x-4)^2=x^2-8x+16 x^2-7x+9=0 D=7^2-4*1*9=49-36=13 x1=(7-√13)/2~1,69<4 - не подходит x2=(7+√13)/2~5,3 - подходит. ответ: (7+√13)/2 2) √(x-3) = √(7x+6) Область определения: -6/7<=x<=3 Это проще, возводим в квадрат x-3 = 7x+6 -9 = 6x x = -1,5 < -6/7 - не подходит ответ: решений нет.
10 9 8 7 6 5 4 3 2 1 = 2016, and
4 4 4 4 4 = 2016
The solutions
First I’d like to say THANK YOU to the hundreds of people who left solutions at the bottom of the question post, on the Guardian Facebook page, on Twitter with the hashtag #MondayPuzzle and in emails to me.
I’ve been totally overwhelmed by the quantity and variety of solutions - and quite embarrassed that my own was so boring!
Evidently there is no unique solution - I was half hoping a computer scientist would let me know exactly how many solutions there are with only the four basic operations. Maybe someone will...
Before we get to my favourites, I’ll explain how I did it myself. Since a fair amount of you were struggling.
The way to solve this type of puzzle is through “enlightened” trial and error. My approach always begins by factorising the year, in this case 2016. Factorising means dividing in to smaller and smaller pieces so that all is left is a string of prime numbers.
2016 breaks down into 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7, which is the same as 25 x 32 x 7
When I saw this I thought that I would leave 7 as it is, and then try to make 25and 32 (32 and 9) out of (10 9 8) and (6 5 4 3 2 1).
If this worked I’d have (10 9 8) x 7 x (6 5 4 3 2 1) = 2016
I quickly noticed that 10 – 9 + 8 = 9. And then playing around that
(6x5) - 4 + 3 + 2 + 1 = 32.
So I had a solution: (10 – 9 + 8) x 7 x ((6x5) – 4 + 3 + 2 + 1) = 2016
It is a pretty dull solution though. The most elegant has to be this one, tweeted earlier by James Annan (and subsequently by others):
(10 x 9 x 8 x 7 x 6)/(5 + 4 + 3 + 2 + 1) = 2016
Область определения:
-3<=x<=7
Возводим всё в квадрат
x+3=4+2√(7-x)+7-x
Оставляем корень отдельно.
2x-8=2√(7-x)
Делим всё на 2
√(7-x)=x-4
Поскольку корень арифметический, то область определения 4 Опять возводим в квадрат
7-x=(x-4)^2=x^2-8x+16
x^2-7x+9=0
D=7^2-4*1*9=49-36=13
x1=(7-√13)/2~1,69<4 - не подходит
x2=(7+√13)/2~5,3 - подходит.
ответ: (7+√13)/2
2) √(x-3) = √(7x+6)
Область определения:
-6/7<=x<=3
Это проще, возводим в квадрат
x-3 = 7x+6
-9 = 6x
x = -1,5 < -6/7 - не подходит
ответ: решений нет.