Go up one from k1, and test the block from there to F, using no more than k-1 drops.

k2 = k1 + 1

Shouldn't that be k2 = k1 -1 ?

This generates the series k1, k1+1, k1+2, ...

which has to sum to F.

Sum: k*(k+1)/2 + k*k*(k+1)/2

= (k+1)*(k+1)/2

so F = (k+1)^2/2

so k = sqrt(2F)

I think that is Sum = k(k+1)^2 over 2

Bob

]]>5^2+3^2=2 (4)^2+2 (x)^2

25+9=2 (16)+2 (x^2)

34=32+2x^2

2x^2=2

x=1

is the below argument logical, or is there an error in the reasoning?

both positive and negative feedback are welcome, but please, keep it classy

Question: What is there?

Reply: Now, there is nothing, something, and something else.

Reasoning:

1.Why? Because if there was something else, we included it.

2.And why is something else included from the beginning? Because it's different from nothing and something, it's...something else.

3.Also nothing is different from something (and both different from something else as mentioned).

I've would do this puzzle in a similar fashion too. The 'solution' uses a sort of averaging effect and I'm a bit suspicious about that in general, although it does seems to work here.

The question actually asks for the time to do a mile so you are one step short of that at the moment.

Bob

]]>I got exactly the same as you did (including the two given questions), and, like you, didn't include numbers with repeated digits (in the spirit of the puzzle).

I initially considered omitting questions whose incorrect copy was also one of the original questions - which would reduce the number of questions (including the two given questions) to 14 - but decided against that because Hungry Horace wouldn't have noticed the 'repeats' as he hadn't recorded the originals.

Here's a link to the puzzle: Sides Reversed Is Puzzle

]]>https://www.mathsisfun.com/puzzles/9-digit-number-solution.html

Please check!

]]>I think your solution is OK. You could argue it's better as it only uses those digits and dots. But Sam Loyd wouldn't have paid out for it. He'd have said "What's a dot matrix printer?"

later edit: actually it does say "add up to", so you'd have to use the minus as a negative sign (there is a subtle difference).

I've noticed that the given solution to some problems is just one of a number of possibilities. The 'solution' page just says "our Solution" so it's implicit that other solutions may also exist.

In the keys puzzle https://www.mathsisfun.com/puzzles/sam- … uzzle.html I couldn't see an analytical way to get an answer so I number crunched it with a program. And got 9 solutions

Bob

]]>Another related idea is Exact Cover. According to Wikipedia:

Finding Pentomino tilings and solving Sudoku are noteworthy examples of exact cover problems. The n queens problem is a slightly generalized exact cover problem.

Below is pseudo code that returns random number based on desired N and dice sides D. It represent shorter version of more detailed steps explained in solution above. But is not solution to Programming problem, since this return actual random number, while solution to programming problem need to return average/expected rolls needed to get random number.

You've got Bd = byy and Ba = byy. I'm hoping you meant Ba = yyy.

Yes, you are correct, that was typo for Ba - I updated my post to fix that.

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