Friday puzzle -- no computers

Re: Friday puzzle -- no computers

Don Orr

No clue on how to solve this but I was taught that the proper way to say 999 in words is nine hundred ninety nine. No "and" or hyphen. Could the Nuns have been wrong ?

British convention?

Alex Y

This puzzle came from a British source, and one commentator described this as the British way of spelling out numbers. Anyone here from the right side of the pond care to comment?

Anyway, for this puzzle, you need to use the convention

(number of hundreds) "hundred" "and" (number representing tens and ones place, hyphenated if two words)

P.S., I learned it with a hyphen, but without the "and".

Okay, then, with computers 

Re: Friday puzzle -- no computers

Larry Barrett

I tried off and on for awhile, so far (without computer) without success.

Here is my reasoning so far; if offbase, or missing something, let me know.

The letter count for the first word must be 3, 4, or 5 corresponding to the letter count for one, two, ..., nine). The middle two words in all cases are "hundred and", so the letter count options for the first three words must be 63, 84, or 105.

The letter counts for the last word must account for all the possibilites - one through twenty, then the twenty-one to twenty-nine, thirty-one to thirty-nine, etc. to ninety-nine. I ended up with 20 possible word-counts, starting with 189. No ah-ha insights so far.

Re: Friday puzzle -- with computers

Larry Barrett

Ok, I found one that meets criteria, and three that are within one (924, 840, 630).

Alex Y

Correct.

But from your subject line, am I to take it that you used the computer to find it?

That's fine, I relaxed the condition. But if so, go back to your previous post; I think you will find that once you got to the 63/84/105 step, you were 95% of the way there!

Re: 

Larry Barrett

I got it this time with help of computer. I used Excel, and used the LEN function to count letters in words. Nothing fancy, but spreadsheet made it easier to see what I was doing. There is likely a more elegant way of getting the answer. I relooked at the scrap of paper I was using yesterday and found that I did have it, but did not realize it at the time.

Solution

Alex Y

Congrats to Larry for getting the answer to this one.

One solution (the brute force way I solved it with a spreadsheet) is to calculate the "word score" of each number from 100-999 and compare it to the number.

But doing it by hand, you notice that each number has "hundred and" in its expansion, with 7 and 3 letters, so the solution has to be a multiple of 21. So now we are down from 900 cases to 43.

Next, you notice that the number of hundreds has 3,4,or 5 digits, so if the number is in the 100s, 200s, or 600s, it is a multiple of 63; if in the 400s, 500s, or 900s a multiple of 84; or in the 300's, 700s or 800s, a multiple of 105. And the multiple in question is the letter count of the number given by the two right-hand digits. This reduces it to 12 cases, if I counted correctly.

Then, it is easy to do in your head. the multiples of 63 that are in the 100s are 2x63=126, but 26 takes more than two letters to spell; and 3x63=189, but 89 can't be spelled with 3 letters.

For the 200s, 4x63=252, but 52 can't be spelled with 4 letters.

For the 300s, 3x105 = 315, but 15 can't be spelled with 3 letters.

...

For the 600s, 10x63 = 630, but 30 doesn't take 10 letters; 11x63=693, and ninety-three is 11 letters!

A few more cases shows you this is the only solution.