On Friday I described how young Albert had been asked to go down to a factory basement and retrieve a rare clock from the company safe. When he got there he saw the following numbers on the safe door….
77 – 49 - 36 – xx
the next number in the sequence will open the safe. What number should Albert use to get at the rare clock?
If you didn’t try to answer the puzzle, have a go now. For everyone else, the solution is after the break.
So…….the solution is simple…the sequence involves multiplying the two numbers together to get the next number….
7×7 =49
4×9= 36
thus the number that Albert was after was 3×6=18
However, quite a few people said that they had two, and sometimes three, solutions. What else did you come up with?
June 22, 2009 at 12:20 am |
Another solution:
Assuming 77 is at position 1, 49 at 2, and 36 at 3, the three points ({1, 77}, {2, 49}, {3, 36}) fit the quadratic polynomial 7.5 x^2-50.5 x+120. The next point would be {4, 38}, so 38 is another answer.
Another way to look at the same answer is to look at the amount each value is different than the preceding:
77-49=28
49-36=13
The next number in the arithmetic sequence 28, 13… is -2, so the next value is 36 – (-2) = 38.
June 22, 2009 at 1:07 am
wow.
June 22, 2009 at 2:03 am
Sorry, but I think both of these answers are pretty lame:
1. you could do this trick for any sequence of three numbers
2. ditto – all you are doing is taking a ’second derivative’ of the three numbers – in this case, -15.
The fact they both come out at 38 is a coincidence, but it doesn’t make the answers right.
June 22, 2009 at 1:06 am |
If we look it as some kind of symmetry, so you don’t always go from left to right, you have 77, and then 7^2 (which is 49). And then since we also have 6^2 (which is 36) in the middle, we should have 66 at the end. That’s the other one that I came up with. (I am guessing many people probably have arrived at this conclusions as well.)
June 22, 2009 at 8:36 pm
Yep, that was the first answer I came up with!
June 22, 2009 at 1:13 am |
anything that’s not 18 is thinking zebras when you hear hoofs beats
June 22, 2009 at 1:24 am |
Another solution:
Considering the complete text of the puzzle spent time talking up roman numerals and specifically mentioning the number 10, it seems reasonable to take the xx as literal roman numerals, xx = 20, giving 77 – 49 – 36 – 20.
June 22, 2009 at 3:18 pm
That is the answer I came up with.
June 22, 2009 at 1:53 am |
If you look at the King James Bible, and the following verses:
Genesis 7-7: And Noah went in, and his sons, and his wife, and his sons’ wives with him, into the ark, because of the waters of the flood.
Genesis 4-9: And the LORD said unto Cain, Where is Abel thy brother? And he said, I know not: Am I my brother’s keeper?
Genesis 3-6: And when the woman saw that the tree was good for food, and that it was pleasant to the eyes, and a tree to be desired to make one wise, she took of the fruit thereof, and did eat, and gave also unto her husband with her; and he did eat.
Then count the number of people mentioned in each verse:
– 7-7: Noah + Wife + 3 Sons (and 3 wives) = 8
– 4-9: Cain + Abel = 2
– 3-6: Woman + Husband = 2
8 + 2 + 2 = 12
Multiply that number of 2 for the number of each animal that Noah had on his ark, and you get the answer:
24
June 22, 2009 at 9:27 am
This answer brought a massive smile to my face.
I have a picture of a theif stood by the safe with a copy of the King James Bible in hand trying to crack the safe!
June 22, 2009 at 9:30 am
…except of course that Noah didn’t have two of each animal – it was 7 of each animal, and 2 of the unclean ones.
However, marvellous reasoning anyway. Can’t believe that wasn’t the right approach.
June 22, 2009 at 2:04 am |
You people are nuts! But I LOVE it!
I only came up with 18 and feel very satisfied about that.
June 22, 2009 at 2:15 am |
Oh dear!
I got 38.
By fitting the three known points into a quadratic polynomial*, and solving for the 4th point.
Epic fail, I guess…
__________
* (15x^2-71x+154)/2
->
x y
0 77
1 49
2 36
3 ?
June 22, 2009 at 2:16 am |
P.S. I just spied the first post, which AGREES with me!
June 22, 2009 at 2:52 am |
i came up with 15. 77-49-36
)
I took subtraction to be the rule for the first digit of each pair and addition for the second digit in each pair but only the last digit.
So 7 – 4 = 3 Therefore 4-3=1 for the first digit.
For the addition, 7+9=16 so 9+6=15 which is 5 for the second digit
So 1 and 5 make 15.
June 22, 2009 at 3:06 am |
Similar to what Richard mentions above: I figured that the “xx” indicated that the next number in the sequence was “20″ – particularly as Albert dealt with roman numerals and would get it, where a thief might not!
June 22, 2009 at 9:05 am
I had the kids mathematics answer ie. seven sevens are fourty nine etc.
XX=20 is a brilliant answer.
Lee Daniel Crocker makes a good point but I believe that a good safe would reset the dials each time you try the door meaning that you would need to dial the first three numbers then each possibility to open the safe. So young Albert’s tea would be cold by the time he came back with the clock.
June 22, 2009 at 3:18 am |
Andrew,
It’s not a coincidence that they come out with the same answer – they are two different ways of taking the same approach – it would be like saying n^2 + n is different from n(n + 1). One of them uses a second-order function directly, and the other uses a linear function of the first derivative. They are both the same approach.
Of course, if you go to higher orders, you can make any number of “next” answers. If I assume the number ‘before’ 77 is 100 and do a third-order fit, I’ll get a different number for the ‘next’ number than if the assumed number ‘before’ 77 were 101. There inherently are an infinite number of answers to a problem like this.
(It’s kinda like a bad question I got at a trivia competition recently: “What did George Washington do in 1796?” Somehow, eat, drink, sleep, etc. were judged to be things he _didn’t_ do that year)
So your complaint that I could do this “trick” for any sequence of numbers is right – that was my point in doing it. 18 was the first number I found by immediate inspection, and 38 was the number I found by realizing that I could use a polynomial fit.
But it doesn’t mean that one answer is legitimate and the other answer is lame (or horses vs. zebras, as scibuff says). They are both mathematically legitimate. And working as an engineer, if I were given these numbers as experimental points and asked to predict the next one, I would definitely use a polynomial fit for extrapolation before going to multiplying digits. Digit multiplication is interesting for number theory, but meaningless for mechanical engineering.
June 22, 2009 at 11:42 pm
aha, yes of course they are the same thing – I should have noticed that.
I still don’t agree that it’s a good answer for a ‘puzzle’ though – this isn’t an engineering course
June 22, 2009 at 3:30 am |
The best idea I had was 77 – 49 – 36 – 66; here, the second number is a product of 7×7, and the third number is a product of 6×6.
June 22, 2009 at 4:01 am |
18 was the obvious answer by simply multiplying. Then comes in our brain trying to seek patterns. This is what else I had:
There seems to be a big dash between 77 & 49 a smaller dash between 49 & 36 and again a big dash between 36 & the mystery number. That suggested some sort of symmetry (big dash, small dash, big dash). If you literally think of the big dash as the minus sign you get 77-49 = 28. The mystery number must be 8 for the symmetry to hold (36-8=28). Of course that’s a bunch of “Da Vinci Code” baloney, but it makes sense in a weird kinda way.
I also had another way of getting to the 18 but I can’t remember it now.
June 22, 2009 at 6:04 am |
My second answer was twenty based on the wording: “When he got there he saw the following numbers on the safe door…. 77 – 49 – 36 – xx”
June 22, 2009 at 6:07 am |
My second “solution” is to simply recognize that every rotary-dial combination lock I’ve ever used (and the picture is of this type) has the property that after you dial all but the last number, you can try the latch at every point on the dial as you move it one notch at a time, so the answer to which last number he should use is simply “whichever one works”. No need to solve any math puzzles at all.
June 22, 2009 at 6:21 am |
The first thing I noticed was that 6*6=36, so my first “answer” was the symmetrical 66. The second kind-of-answer I came up with was 25: maybe 77 is a kind of initial setting, the next number is 7*7, the next one 6*6, and the final 5*5. And only after that I saw the perfectly simple correct solution. Duh. Keep it simple, stupid…
June 22, 2009 at 6:25 am |
It’s disappointing but not particularly to see that the “second solutions” people had were really contrived and not “a-ha!” moments at all.
June 22, 2009 at 6:30 am |
Official answer as expected following the logic that this was a ‘puzzle’ not a maths exam. I do like the XX=20 as an alternative answer for a puzzle though, I certainly missed that. Other maths based answers were clearly possible, but ruled out as unlikely in the context.
June 22, 2009 at 6:50 am |
I came up with 66. I like symmetry.
June 22, 2009 at 8:19 am |
I came up with 38 and 66, but I suspected none of them was correct. When I eventually saw the pattern that led to 18, it seemed obvious that this was the correct answer.
June 23, 2009 at 10:59 am
I went with 18, because it seemed obvious that this was the correct answer – but 20 was also quite an ingenious solution. Interesting to speculate on precisely why some answers seem obviously correct, while others might technically work, but don’t produce that “aha” moment when they are hit upon. If anyone has any views on this, I’d be interested in hearing them.
June 22, 2009 at 8:27 am |
Yep. Wrong again. I went for the symmetry and came up with 66.
June 22, 2009 at 9:31 am |
Another 66er.
June 22, 2009 at 11:28 am |
I got 18 and I was so ridiculously happy about it I had to jump up and down a few times.
June 22, 2009 at 12:59 pm |
Ahhh…i must admitt, that i didn’t see that!…got so lost in another pattern that, in the end i had at least 3 possibilities due to not knowing the real rythm of the sequence…
Here goes (i always was send out of math-class, probably for a good reason)
77, difference -28, =49
49, difference -13, =36
28, difference 13 = sequence 13+2 upwards and
. 13-2 downwards Answer= 25
28, difference 13 = sequence 13+2 upwards and
. 13+2 downwards Answer= 21
or 13+2
. 13+1 Answer =22 , or 13-1 answer=24
And so on….
(truly hate my former math-teach, he said ‘you left your logic at home!’…can i use your couch for an hour PRof Wiseman? haha)
June 22, 2009 at 11:55 pm
sorry, what the hell?
June 23, 2009 at 7:23 am
yes, what the hell…good question
June 22, 2009 at 4:27 pm |
The solution I came was:
77-49-36=-8
So the next number in the sequence “77 – 49 – 36 – ” is “8″
June 22, 2009 at 4:29 pm |
The solution I came to was:
77-49-36=-8
So the next number in the sequence “77 – 49 – 36 – ” is “8″
June 22, 2009 at 11:26 pm |
My first solution was a bit far-reached:
I took the digits in 77 and averaged them, taking the integer of the result.
This comes down to 7. 7^2 = 49
Same goes for 49
49 -> (4+9)/2-> 6.5-> (rounded) 6-> 6^2 -> 36
so for 36
36 -> (3+6)/2 ->4.5 -> (rounded) 4-> 4^2-> 16
16!!!!
Then I the “classic” answer….
June 22, 2009 at 11:54 pm
I think you need to truncate rather than round!
June 23, 2009 at 12:50 am |
[...] Answer to the Friday puzzle! On Friday I described how young Albert had been asked to go down to a factory basement and retrieve a rare clock from [...] [...]
June 23, 2009 at 7:30 am |
sorry, what the…round truncate? lol
June 23, 2009 at 10:38 am |
Interestingly 77 is the only two-digit number that can cope with three round of digit multiplying whilst remaining 2 digits long.
June 23, 2009 at 5:47 pm
Nice observation!
I did some playing with sums of squares of digits when I was younger. There are a few interesting patterns with that.
June 23, 2009 at 1:04 pm |
There are a number of different answers.
I got the one in the puzzle, but I also noticed that he used a different character for the dash between 49 and 36:
77 – 49 – 36 – xx
I still don’t understand why, but here’s one thing I thought of:
The smaller dash is in place of an equal sign:
77-49 = 36-x
77-49-36 = -x
-8 = -x
x = 8
(I didn’t do it that way in my head; I just made the difference between 77 and 49 the same as the difference between 36 and x, but it works out the same way.)
I see people got this answer in other ways, without having to convert it to an equal sign. So I’m calling this one the “correct” answer. :^P
June 24, 2009 at 5:02 pm |
I got 18 at the first glance. But yea, there r always another perspectives which u cn look things. n i am impressed by those who come up with different, various possible solutions. ^_^
June 25, 2009 at 8:08 am |
The answer is 98.
June 25, 2009 at 8:14 am |
ooops… i mean to say answer is 18.
June 25, 2009 at 12:22 pm |
Proud to say I came up with 18 almost straight away, but didn’t stop to consider other solutions as I popped over to see the answer
June 26, 2009 at 5:14 am |
‘the solution is after the break’
the break is 00
the nunber after the break is 01
so, the answer is:
77 49 36 01
June 26, 2009 at 9:23 am |
A little late with replying but I did get it right, 18!
July 3, 2009 at 4:18 am |
i got 18 too. i thought 66 and 8 were brilliant. I TAKE MY HAT OFF TO ALL YALL. Haven’t been challenged like this in a while.I feel unworthy for a prior EE major.
July 4, 2009 at 4:48 pm |
The answer is 42.
Irrespective of the problem.