119
votes
114
votes
Accepted
80
votes
Accepted
Nine gangsters and a gold bar
18 pieces:
In 9 parts:
In 8 parts:
In 7 parts:
We need to be able to split it into 9 parts of 56, so it can't hurt to make 9 pieces of 56 and then split those further. Since we need to do better ...
- 33.6k
76
votes
64
votes
52
votes
Accepted
Make 5 5 5 5 = 19
Here's one way I found:
Or, using just the characters explicitly allowed in the question:
- 14.4k
46
votes
Accepted
The largest Monday number
Notice 9867312 is a Monday number.
The largest Monday number may not contain 5 because in this case it would end in 5, and thus not be divisible by 2, 4 and 8, so it would have at most 6 digits.
On ...
- 2,537
43
votes
42
votes
Accepted
First digit of 3^2020
I think it's a
Explanation:
According to my calculator (this answer was posted before the no-computers tag was added),
If we're not allowed to use computers, I would
The same trick is used to ...
- 28k
42
votes
x⌊x⌊x⌊x⌋⌋⌋ = 2020
Observations to give lower and upper bounds:
So we know for sure
Now the whole thing becomes
Contradiction ... and now I realise my implicit assumption that
Going back to those two observations at ...
- 116k
39
votes
37
votes
Make 5 5 5 5 = 19
If the double factorial is allowed, then I propose
WolframAlpha agrees that the result is 19.
37
votes
Accepted
36
votes
First digit of 3^2020
This is one way to do it with the help of binary numbers. It is called exponentiation by squaring.
EDIT
Thanks to the few comments that pointed out the mistakes in my calculations. I was very lucky to ...
- 937
34
votes
Accepted
33
votes
31
votes
Accepted
Professor Halfbrain and the sum of the digits of all divisors
Professor Halfbrain's theorem is
Proof
- 132k
31
votes
Make 5 5 5 5 = 19
I am quite sure it is not the expected answer but it is the immediate answer comes into my mind.
- 321
30
votes
30
votes
Number of divisors equals square root
The number of positive divisors is a multiplicative function in the number theoretical sense. The square root is also multiplicative. We can therefore proceed prime factor by prime factor. If ...
- 18.2k
29
votes
29
votes
Accepted
What makes this polynomial a square number?
Of course, $x=0$ is an answer, so let's look for non-zero ones from now on. If the given expression is a perfect square, so is
Now we try to estimate it by "nearby" perfect squares.
One ...
- 18.3k
27
votes
27
votes
Accepted
25
votes
Nine gangsters and a gold bar
It has been shown that fewer than 16 pieces is not possible (due to the 8-way split then requiring someone to have a single piece larger than a single portion of the 9-way split). I'll show by ...
- 1,134
25
votes
Nine gangsters and a gold bar
22 pieces
Suppose the weight of the bar is 504. I chose 504 since 7*8*9=504, so, the numbers would be easy to work with.
7 pieces of weight 56 (A)
8 pieces of weight 7 (B)
6 pieces of weight 9 (C)
...
- 4,020
25
votes
Accepted
25
votes
Accepted
What number follows up next? Part 2
OEIS doesn't list this sequence. After analyzing the pattern, I come to the conclusion that (one) answer is
Explanation:
EDIT (05/06/18): I submitted this sequence to OEIS and it has (finally) been ...
- 1,360
25
votes
Nice no-computers way to find limerick primes?
Obviously a is one of {1,3,7,9} and a,b are coprime. Also, b can't be a multiple of 3 regardless (else our number is a multiple of 3). That leaves 23 possibilities (4 for a, 6 for b, but we can't have ...
- 116k
24
votes
Accepted
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