For general N>6, the following solution works.

The number of 1?s in this paragraph is N-3; the number of 2?s is 3; the number of 3?s is 2; .. the number of N-3?s is 2. All other blanks are 1.

This solution works as long as N-3 > 3, i.e., N > 6.

Thought process

Again, the sum of the blanks has to be 2N. That implies the average in a blank is 2. This must mean most values are small.

Now, most blanks are small imply the majority of the values are 1 or 2. Most blanks cannot have 2 because then we won’t be able to satisfy all the statements ‘The numbers of K’s in the paragraph are 2’.

So, let us assume that most blanks are 1 and try possible values for the first blank. Suppose we fill the first blank with N-p. Then the (N-p)th blank will have value at least 2. A larger value will again lead to all constraints not being satisfied. So, the (N-p)th blank is 2.

Now, the second blank is at least 2. But it cannot be 2 (because then there would be three 2’s). So, the second blank is greater than equal to 3. As it turns out, it cannot be greater. So, this blank is filled with 3.

Now, the third blank can be filled with 2 and all others with 1. And the value of p can be determined by the fact that the sum is 2N. Comes out to be 3.

I agree the proof is not rigorous about why no other solution works.

]]>All leading zeros;

Trailing zeros when they are merely placeholders to indicate the scale of the number

Spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.

e.g., For A.D. 2009, all digits are significant, but for May 05, the 0 is not significant.

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An escaped prisoner finds himself in the middle of a square swimming pool. The guard that is chasing him is at one of the corners of the pool. The guard can run faster than the prisoner can swim. The prisoner can run faster than the guard can run. The guard does not swim. Which direction should the prisoner swim in in order to maximize the likelihood that he will get away?

]]>are they numerical representations of wavelength and frequency?

or are they resonant frequency math expressions?

for example what does 1.3333 mean in terms of the real world….

how does pi relate to harmonic numbers?

sincerely,

math lover

]]>the 3rd and 4th rounds work fine even for groups of a single coin, because you can group the comparisons like this:

Round 3:

1 3

2 4

5 7

6 8

…

Round 4:

3 5

4 6

7 9

8 10

…

[/spoiler]

The last question you ask is interesting, it’s like sorting algorithms where all the comparisons need to be specified ahead of time: Sorting Networks.

But I think it has a rather different answer… [spoiler]

You need to compare every possible pair of coins.

If there are any two coins A and B which are not compared with each other, then suppose all other coins (call them “class C coins”) are equal to each other but unequal to A or B. In this case you cannot know whether A and B are the same or not.

[/spoiler]

Would it help if you are told there is at least one of each type of coin?