14



12

OH MY GOD I just got it. Like, as I was looking at my own question. Boy do I feel silly. The next number is PHEW that was bugging the bejeezus out of me! Hahaha


12

I claim that no arrangement is possible in the case with cards ace through four. Note that _A_A_A_A_ (_ is any other card) must be part of the arrangement. Now consider two cases: the row starts and ends with aces - this is obviously not possible, since that arrangement would only hold 7 cards, and we require 16. there are cards to the left and right ...


10

Are the next 3 terms The numbers are So: Continuing the sequence: The same pattern applies for the second hint, and the third


7

A similar answer to Quintec's (asserting that it is impossible): Consider that you only have 12 cards that are not a 4, and that you need 12 total cards between the 4s. So the entire arrangement must look like 4 _ _ _ _ 4 _ _ _ _ 4 _ _ _ _ 4 As per Quintec's observation, somewhere in there we must have _ A _ A _ A _ A _ But you can see that there ...


6

As claimed above by other users, the arrangement is imposible with four sets of four cards. The earliest solution occurs for four sets of cards 1 to 24, for which there are three solutions. These are known as Langford Quads and are described at John Miller's excellent blog on Langford's Problem. These are Richard Noble's three solutions.


6

The second hint was helpful. Once you see things that way, it is not hard to see the relationship between the left and right side of the equality sign. The answer:


5



5



5

The answer to this is: Since:


4

Since our dear friend balazs.com has not deigned to provide a comprehensible explanation to us lowly mortals, I've provided a simple explanation here: The missing number is: This is because:


4

The answer is: because:


4

The next term is: Because the terms are all: Meaning that the next term must be:


3

because even terms reduce by 3.5


3

The solution is:


3

Solution: Finding the Solution: This gets us: We now have: This gives us: Now, we have: Here we have: In conclusion:


3

An answer for 5s (updated)


3

Reason:


3

second: because: third: because:


2



2

Mohiri got the correct answer, but they came to that answer for a slightly different reason than the logic I was using. I'm adding this answer to clarify my approach.


2

Possible Answers First one Second one Third one Fourth one (spotted by @Stiv)


1

The sequence is two sequences interlaced. That is what makes it difficult to figure out.


1

My answer is X=4 and 5341068+5. Sketch is on photo below.


1

The first sequence So the next term is


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