119
votes
114
votes
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76
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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 ...
- 27.9k
42
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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
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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
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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
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33
votes
31
votes
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Professor Halfbrain and the sum of the digits of all divisors
Professor Halfbrain's theorem is
Proof
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31
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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
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30
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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 ...
- 19.1k
29
votes
29
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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 ...
- 18.6k
27
votes
27
votes
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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
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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 ...
- 117k
24
votes
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24
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Consecutive integers around a circle
I have a solution using 448 consecutive integers.
The integers are centered on a number which I will call $c$, which is:
...
- 5,451
24
votes
24
votes
Accepted
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