Puzzles with numbers and things

[snookersfun]

Hi all! I have had perfect answers to questions 138 and 139 from davis_greatest and from Abextra. Well done and more fame for you both!

So, until one of them will put up the answers the rounds can still be solved by anyone feeling up to it and having the time!

[snookersfun]

Round 140

Meanwhile here is another nice one for all of you:

Round 140:

An absent-minded math professor was telling his eccentric colleague about losing some of his paperwork.
"I can't find my special number lists," he complained.
"What was so special about them?" replied his colleague.
"Well," he explained, "I call a list of numbers 'special' if it has the following 7 properties:"
(1) All of the numbers on the list are positive integers, base 10.
(2) There is more than one number on the list.
(3) All of the numbers on the list have the same number of digits.
(4) No two of the numbers on the list are equal.
(5) The smallest number on the list is equal to the total number of digits of all the numbers on the list.
(6) The largest number on the list is equal to the sum of all of the digits of all of the numbers on the list.
(7) The number of items on the list is equal to the largest number divided by the smallest number.
"There must be thousands of special lists," replied his colleague.
"Hmm, I'm not sure about that, but I was only interested in four of them."
"What four?"
"What for? Well, I divide the sum of the numbers on each list by 100 and the four remainders comprise the four-number combination to my office safe. The safe is locked and I can't remember the combination without them!"
"No, I meant which four!"
"Well obviously I can't remember the lists, but I do recall their properties. First, all of the numbers on a list that lie strictly between the smallest and largest numbers I will call the middle-numbers. Now all of the middle-numbers on the first list are primes, and all of the middle-numbers on the second list are composite. Furthermore, the two largest middle-numbers on the third list do not appear on the first or second lists. Finally, the largest middle-number on the fourth list does not appear on any of the other three lists, and the smallest middle-number of the fourth list is not equal to the smallest middle-number of any of the other three lists."
"Hmm, let me think about that. By the way, what do you have in your safe?"
"I can't remember! But I usually lock up important papers that I am currently working on."
The next day the eccentric professor approached the absent-minded professor while waving a handful of papers. "I found your lists!" he shouted.
"Where in the world did you find them? I've looked everywhere!"
"I found them in your safe!"

Questions:
(1) What is the combination to the safe?
(2) How many possible "special" lists are there?

[abextra]

moving on to the next round:

Round 138:

Archeologists recently made an amazing find in a new dig site not far from Rome. It was an approximately two thousand year old stone tablet on which was engraved what could only be described as a primitive cross-number diagram. It had only four numbers, two horizontal (reading left to right) and two vertical (reading top to bottom), as shown here in the blank diagram with the numbers removed. All four of the numbers were different and all were less than four thousand. The bottom horizontal one was the smallest. After consulting with mathematicians they discovered that all four of the numbers were squares, but only the smallest number was a fourth power. Also the sum of the four numbers was a prime. Using only the information given here can you find the unique original diagram?

Answers initially by PM please

A tip concerning this one, I have just discovered that excel does have a ROMAN function, which should make live a lot easier!

Here are some square numbers, which can be written in 5-letter Roman numerals:

.81 - LXXXI --------- 441 - CDXLI --------- 1521 - MDXXI
144 - CXLIV --------- 529 - DXXIX --------- 2025 - MMXXV
169 - CLXIX --------- 625 - DCXXV --------2401 - MMCDI
196 - CXCVI --------- 961 - CMLXI --------- 2601 - MMDCI
225 - CCXXV --------1024 - MXXIV----------3600 - MMMDC
256 - CCLVI -------- 1156 - MCLVI
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ,,,,,,,,(bold - fourth power)

Maybe someone likes to have a go!

[abextra]

Meanwhile here is another nice one for all of you:

Round 140:

An absent-minded math professor was telling his eccentric colleague about losing some of his paperwork.
"I can't find my special number lists," he complained.
"What was so special about them?" replied his colleague.
"Well," he explained, "I call a list of numbers 'special' if it has the following 7 properties:"
(1) All of the numbers on the list are positive integers, base 10.
(2) There is more than one number on the list.
(3) All of the numbers on the list have the same number of digits.
(4) No two of the numbers on the list are equal.
(5) The smallest number on the list is equal to the total number of digits of all the numbers on the list.
(6) The largest number on the list is equal to the sum of all of the digits of all of the numbers on the list.
(7) The number of items on the list is equal to the largest number divided by the smallest number.
"There must be thousands of special lists," replied his colleague.
"Hmm, I'm not sure about that, but I was only interested in four of them."
"What four?"
"What for? Well, I divide the sum of the numbers on each list by 100 and the four remainders comprise the four-number combination to my office safe. The safe is locked and I can't remember the combination without them!"
"No, I meant which four!"
"Well obviously I can't remember the lists, but I do recall their properties. First, all of the numbers on a list that lie strictly between the smallest and largest numbers I will call the middle-numbers. Now all of the middle-numbers on the first list are primes, and all of the middle-numbers on the second list are composite. Furthermore, the two largest middle-numbers on the third list do not appear on the first or second lists. Finally, the largest middle-number on the fourth list does not appear on any of the other three lists, and the smallest middle-number of the fourth list is not equal to the smallest middle-number of any of the other three lists."
"Hmm, let me think about that. By the way, what do you have in your safe?"
"I can't remember! But I usually lock up important papers that I am currently working on."
The next day the eccentric professor approached the absent-minded professor while waving a handful of papers. "I found your lists!" he shouted.
"Where in the world did you find them? I've looked everywhere!"
"I found them in your safe!"

Questions:
(1) What is the combination to the safe?
(2) How many possible "special" lists are there?

On the second list there could be numbers

14, 69, 78, 87, 88, 96, 98.

They are all positive integers, all of them have two digits, in total 14 digits on the list. All of the middle numbers are composite.

The sum of digits 1 + 4 + 6 + 9 + 7 + 8 + 8 + 7 + 8 + 8 + 9 + 6 + 9 + 8 = 98

The largest number 98 divided by the smallest number 14 is 7 and there are exactly 7 numbers on the list. All of the middle numbers are composite.

The sum of numbers 14 + 69 + 78 + 87 + 88 + 96 + 98 = 530

530 divided by 100 gives a reminder 3.

So far so good. I think it's possible to do lists 3 and 4, with the same smallest and largest number and the same sum of numbers... But I have no ideas about list 1!

[snookersfun]

On the second list there could be numbers

14, 69, 78, 87, 88, 96, 98.

They are all positive integers, all of them have two digits, in total 14 digits on the list. All of the middle numbers are composite.

The sum of digits 1 + 4 + 6 + 9 + 7 + 8 + 8 + 7 + 8 + 8 + 9 + 6 + 9 + 8 = 98

The largest number 98 divided by the smallest number 14 is 7 and there are exactly 7 numbers on the list. All of the middle numbers are composite.

The sum of numbers 14 + 69 + 78 + 87 + 88 + 96 + 98 = 530

530 divided by 100 gives a reminder 3.

So far so good. I think it's possible to do lists 3 and 4, with the same smallest and largest number and the same sum of numbers... But I have no ideas about list 1!

So far so good Abextra, list 2 is fine, except that the remainder would be 30.
I would have thought list one was one of the easiest (again same smallest and largest number and the same sum of numbers. Note, just the middle numbers have to fulfill the condition of prime numbers!).

[abextra]

Well, I divide the sum of the numbers on each list by 100 and the four remainders comprise the four-number combination to my office safe.

I'm afraid I don't understand the remainder thing. I better leave this question...

[snookersfun]

I'm afraid I don't understand the remainder thing. I better leave this question...

530:100=5 (remainder 30), so 30 is the second of the 4 numbers for the combination of the safe
But never mind that part, as long as anyone comes up with the lists, that would be fine as well

[davis_greatest]

530:100=5 (remainder 30), so 30 is the second of the 4 numbers for the combination of the safe
But never mind that part, as long as anyone comes up with the lists, that would be fine as well

So the 4 remainders are not necessarily single-digits? So the combination to the safe can be more than 4 digits long? (That must be difficult to open then! This question is reminding me of long, long ago at school, when I used to bet my friends that I could open their combination briefcases in about 30 seconds. )

[snookersfun]

So the 4 remainders are not necessarily single-digits? So the combination to the safe can be more than 4 digits long? (That must be difficult to open then! This question is reminding me of long, long ago at school, when I used to bet my friends that I could open their combination briefcases in about 30 seconds. )

Yes, we have to find four numbers (0-99 obviously) not four digits
lol, about the briefcases, my kids are doing this with today's improved locker-locks. I actually have one in front of me at the moment and it has numbers from 0-39 on it (although don't blame me for the storyline of this question, I am rather untalented when it comes to making up these stories, so I just copied it).

[snookersfun]

Breaking news: Abextra has now found the combination to the safe.
Well done, braving the unruly remainders and all!!!
She is also pretty much on her way to find the number of possible lists. But there is still a chance for the seemingly hibernating (or extremely busy, OK,OK!) rest of you to catch up and find the answers to this puzzle. So, until Abextra will put up her full solution, have fun

Meanwhile here is another nice one for all of you:

Round 140:

An absent-minded math professor was telling his eccentric colleague about losing some of his paperwork.
"I can't find my special number lists," he complained.
"What was so special about them?" replied his colleague.
"Well," he explained, "I call a list of numbers 'special' if it has the following 7 properties:"
(1) All of the numbers on the list are positive integers, base 10.
(2) There is more than one number on the list.
(3) All of the numbers on the list have the same number of digits.
(4) No two of the numbers on the list are equal.
(5) The smallest number on the list is equal to the total number of digits of all the numbers on the list.
(6) The largest number on the list is equal to the sum of all of the digits of all of the numbers on the list.
(7) The number of items on the list is equal to the largest number divided by the smallest number.
"There must be thousands of special lists," replied his colleague.
"Hmm, I'm not sure about that, but I was only interested in four of them."
"What four?"
"What for? Well, I divide the sum of the numbers on each list by 100 and the four remainders comprise the four-number combination to my office safe. The safe is locked and I can't remember the combination without them!"
"No, I meant which four!"
"Well obviously I can't remember the lists, but I do recall their properties. First, all of the numbers on a list that lie strictly between the smallest and largest numbers I will call the middle-numbers. Now all of the middle-numbers on the first list are primes, and all of the middle-numbers on the second list are composite. Furthermore, the two largest middle-numbers on the third list do not appear on the first or second lists. Finally, the largest middle-number on the fourth list does not appear on any of the other three lists, and the smallest middle-number of the fourth list is not equal to the smallest middle-number of any of the other three lists."
"Hmm, let me think about that. By the way, what do you have in your safe?"
"I can't remember! But I usually lock up important papers that I am currently working on."
The next day the eccentric professor approached the absent-minded professor while waving a handful of papers. "I found your lists!" he shouted.
"Where in the world did you find them? I've looked everywhere!"
"I found them in your safe!"

Questions:
(1) What is the combination to the safe?
(2) How many possible "special" lists are there?

[davis_greatest]

Round 141 - Win, lose or draw

So, we are back up?! Go on then, let's hear the answer to the safe combination question. I'm going to get back to posting questions...

Here, then, copied across from the temporary (yrush) home:

My pet apes Charlie, Gordon and Oliver are (a) very intelligent and (b) trusting, but (c) they like to tease the other apes in a good-natured way. This means (a) that they are perfect at logic, and know that the other apes are too, (b) will believe everything they hear (unless they know that it must be untrue), but (c) they may on occasion tell lies to tease the others. As they are good-natured, they will only tell lies if they know that the others will know that they must be teasing.

1) “I was there at the draw,” said Charlie. “This is what they did. They got that woman who did such a brilliant* job on Radio 5 Live last year to put the names of the 16 seeds and 16 qualifiers all into the same hat, and then she drew out 2 names at a time at random until all 32 had been drawn. If both names were seeds, they put them into a “pile of seeds”, and if both were qualifiers, they put them into a “pile of qualifiers”. If one name was a seed and the other a qualifier, they paired them up.

Anyway, at the end, they found that they had 4 names in the seeds pile and 6 in the qualifiers pile, so they couldn’t match them up, so they put all 32 names back into the hat and started the whole thing again. On the second attempt, after drawing all 32 names, they had 4 names in the seeds pile and 2 in the qualifiers pile, so again they put all 32 names back into the hat and started the whole thing again.

At the end of the third go, they found that they had 2 names in the seeds pile and also 2 in the qualifiers pile. The other 14 seeds and 14 qualifiers had been paired up. So, they took one of the seeds at random from the pile of 2 seeds – it was Ronnie O’Sullivan – and one from the pile of 2 qualifiers – it was Ding Junhui; and they paired them up. The remaining seed and qualifier were Steve Davis and John Parrott respectively. And that, my ape friends, is how they did the draw.”

*this is sarcasm, not teasing – did I mention that my apes can be sarcastic?

2) “I don’t believe you,” said Gordon.

3) “And I don’t believe you,” said Oliver to Gordon.

4) “I believe exactly one of the last two statements I have heard,” said Charlie to Gordon and Oliver.

5) “I believe exactly one of the last two statements that I have heard,” said Gordon.

6) “Of the 6 things we have said, including what I am saying now,” said Oliver, “more include a lie than the number that are entirely true.”

Explain whether each statement (1 to 6) is true or untrue.

I know that I stopped awarding points, but you can have:

one point for a correct solution; plus
one bonus point if you remain less confused than I became while typing all that (especially trying to formulate statement 6).

[snookersfun]

d_g good of you to copy that; I don't really fancy to move back and forth now this here seems to be running.

In my PMs I did find a message from Abextra, who had solved the safe puzzle completely now (very well done!!!!) before the 'big crash'. So, Abextra, will you please put up the solution

[abextra]

Round 140:

An absent-minded math professor was telling his eccentric colleague about losing some of his paperwork.
"I can't find my special number lists," he complained.
"What was so special about them?" replied his colleague.
"Well," he explained, "I call a list of numbers 'special' if it has the following 7 properties:"
(1) All of the numbers on the list are positive integers, base 10.
(2) There is more than one number on the list.
(3) All of the numbers on the list have the same number of digits.
(4) No two of the numbers on the list are equal.
(5) The smallest number on the list is equal to the total number of digits of all the numbers on the list.
(6) The largest number on the list is equal to the sum of all of the digits of all of the numbers on the list.
(7) The number of items on the list is equal to the largest number divided by the smallest number.
"There must be thousands of special lists," replied his colleague.
"Hmm, I'm not sure about that, but I was only interested in four of them."
"What four?"
"What for? Well, I divide the sum of the numbers on each list by 100 and the four remainders comprise the four-number combination to my office safe. The safe is locked and I can't remember the combination without them!"
"No, I meant which four!"
"Well obviously I can't remember the lists, but I do recall their properties. First, all of the numbers on a list that lie strictly between the smallest and largest numbers I will call the middle-numbers. Now all of the middle-numbers on the first list are primes, and all of the middle-numbers on the second list are composite. Furthermore, the two largest middle-numbers on the third list do not appear on the first or second lists. Finally, the largest middle-number on the fourth list does not appear on any of the other three lists, and the smallest middle-number of the fourth list is not equal to the smallest middle-number of any of the other three lists."
"Hmm, let me think about that. By the way, what do you have in your safe?"
"I can't remember! But I usually lock up important papers that I am currently working on."
The next day the eccentric professor approached the absent-minded professor while waving a handful of papers. "I found your lists!" he shouted.
"Where in the world did you find them? I've looked everywhere!"
"I found them in your safe!"

Questions:
(1) What is the combination to the safe?
(2) How many possible "special" lists are there?

1. What is the combination to the safe?

The combination is 03 - 30 - 57 - 39

List 1 : 14, 59, 67, 79, 89, 97, 98 , sum 503
List 2 : 14, 69, 78, 87, 88, 96, 98 , sum 530
List 3 : 14, 79, 88, 89, 94, 95, 98 , sum 557
List 4 : 14, 78, 79, 88, 89, 93, 98 , sum 539

2. How many possible "special" lists are there?

On these lists the digital sums of the middle numbers must add up to 76, so that if we add the digital sums of 14 and 98, the total sum will be 98.

Between 14 and 98 there are

8 numbers with a digital sum of 11 - 29, 38, 47, 56, 65, 74, 83, 92
7 . . . . . . . . . . . . . . . . . . . . 12 - 39, 48, 57, 66, 75, 84, 93
6 . . . . . . . . . . . . . . . . . . . . 13 - 49, 58, 67, 76, 85, 94
5 . . . . . . . . . . . . . . . . . . . . 14 - 59, 68, 77, 86, 95
4 . . . . . . . . . . . . . . . . . . . . 15 - 69, 78, 87, 96
3 . . . . . . . . . . . . . . . . . . . . 16 - 79, 88, 97
1 . . . . . . . . . . . . . . . . . . . . 17 - 89

I found a formula n!/k!(n-k)! (someone smarter please explain it ) and 10 different combinations of these digital sums :

17, 16, 16, 16, 11 - 8 lists
17, 16, 16, 15, 12 - 84lists
17, 16, 16, 14, 13 - 90 lists
17, 16, 15, 15, 13 - 108 lists
17, 16, 15, 14, 14 - 120 lists
17, 15, 15, 15, 14 - 20 lists
16, 16, 16, 15, 13 - 24 lists
16, 16, 16, 14, 14 - 10 lists
16, 16, 15, 15, 14 - 90 lists
16, 15, 15, 15, 15 - 3 lists

So in total there are 557 different ''special'' lists.

Big thanks to Snookersfun and davis_greatest for help and encouragement, , I would have given up without that!

[davis_greatest]

Round 141 - Win, lose or draw update

Congratulations to snookersfun for correctly solving round 141, and to abextra who hasn't yet decided whether Charlie's first statement was true or false but has given alternative answers for whether statements 2 to 6 are true or false, depending on whether statement 1 is!

Anyone who wants to post an answer, please do so - in the 2, 3 or 4 hours, after which time I expect to be back and ask another question...

[davis_greatest]

Closing round 141...

OK...

1) Charlie's first story about the draw is manifestly teasing (a lie). This is because we started with the same number of seeds as qualifiers, and each time a seed is drawn with a qualifier, they are paired up, so at the end there must be the same number of names in the seeds pile as in the qualifier pile. Therefore, Charlie's assertion that on the first two attempts, there were different numbers in each pile must have been codswallop.

2) “I don’t believe you,” said Gordon.
Gordon is telling the truth because his perfect logic tells him that Charlie was lying.

3) “And I don’t believe you,” said Oliver to Gordon.
Oliver is lying because he knows that Gordon was telling the truth when saying that he did not believe Charlie.

4) “I believe exactly one of the last two statements I have heard,” said Charlie to Gordon and Oliver.
Charlie is telling the truth - he knows that statement 2 was true and 3 was untrue.

5) “I believe exactly one of the last two statements that I have heard,” said Gordon.
Gordon is telling the truth - he knows that statement 4 was true and 3 was untrue.

6) “Of the 6 things we have said, including what I am saying now,” said Oliver, “more include a lie than the number that are entirely true.”
So far, we had 3 truths and 2 lies. If statement 6 were true, there would be 4 truths and 2 lies, contradicting statement 6, so it cannot be true!. If statement 6 is a lie, there would be 3 truths and 3 lies, meaning that statement 6 is still a lie - which it is!

Congratulations again to snookersfun; semi-congratulations to abextra... and next round to follow shortly...

So, we are back up?! Go on then, let's hear the answer to the safe combination question. I'm going to get back to posting questions...

Here, then, copied across from the temporary (yrush) home:

My pet apes Charlie, Gordon and Oliver are (a) very intelligent and (b) trusting, but (c) they like to tease the other apes in a good-natured way. This means (a) that they are perfect at logic, and know that the other apes are too, (b) will believe everything they hear (unless they know that it must be untrue), but (c) they may on occasion tell lies to tease the others. As they are good-natured, they will only tell lies if they know that the others will know that they must be teasing.

1) “I was there at the draw,” said Charlie. “This is what they did. They got that woman who did such a brilliant* job on Radio 5 Live last year to put the names of the 16 seeds and 16 qualifiers all into the same hat, and then she drew out 2 names at a time at random until all 32 had been drawn. If both names were seeds, they put them into a “pile of seeds”, and if both were qualifiers, they put them into a “pile of qualifiers”. If one name was a seed and the other a qualifier, they paired them up.

Anyway, at the end, they found that they had 4 names in the seeds pile and 6 in the qualifiers pile, so they couldn’t match them up, so they put all 32 names back into the hat and started the whole thing again. On the second attempt, after drawing all 32 names, they had 4 names in the seeds pile and 2 in the qualifiers pile, so again they put all 32 names back into the hat and started the whole thing again.

At the end of the third go, they found that they had 2 names in the seeds pile and also 2 in the qualifiers pile. The other 14 seeds and 14 qualifiers had been paired up. So, they took one of the seeds at random from the pile of 2 seeds – it was Ronnie O’Sullivan – and one from the pile of 2 qualifiers – it was Ding Junhui; and they paired them up. The remaining seed and qualifier were Steve Davis and John Parrott respectively. And that, my ape friends, is how they did the draw.”

*this is sarcasm, not teasing – did I mention that my apes can be sarcastic?

2) “I don’t believe you,” said Gordon.

3) “And I don’t believe you,” said Oliver to Gordon.

4) “I believe exactly one of the last two statements I have heard,” said Charlie to Gordon and Oliver.

5) “I believe exactly one of the last two statements that I have heard,” said Gordon.

6) “Of the 6 things we have said, including what I am saying now,” said Oliver, “more include a lie than the number that are entirely true.”

Explain whether each statement (1 to 6) is true or untrue.

I know that I stopped awarding points, but you can have:

one point for a correct solution; plus
one bonus point if you remain less confused than I became while typing all that (especially trying to formulate statement 6).