Puzzles 2

These puzzles all required me to spend some time. For some, I needed to think a lot -- I typically spent my commute time thinking about it. Sometimes it needed thinking both going and coming to work and sometimes a lot more. For some problems, I needed paper and pen and sometimes a computer program helped. Although some could probably be solved using a "brute-force" computer program, I tried not to do all the work on a computer, figuring out some of the basic mathematics before using a computer application -- typically a spreadsheet program is all that I used.
My three children's ages Solution
I was having coffee at Starbucks with my nephew Kartik the other day. He looked up at the door and a grin lit up his face. A woman, about his age, was walking in. He stood up, completely ignoring the fact that I was in the middle of a sentence, and was at her side in a bound.

They met like old friends (from their college days, it turns out, where they sat next to each other in Math and Logic classes) and after they had declaimed loudly about how l-o-n-g it was since they had last met, and had exchanged Facebook IDs, he asked her the inevitable question:
"Are you married?"
"Yes. You?"
"Until recently, I was," he said. "Kids?"
"Yes, I have three."
"Oh?" he raised his eyebrows. "I am an uncle to three people? How old are they?"
A mischievous look entered her eyes. "The product of their ages, as integers, is 36."
"Hmm." He frowned. "You have to give me more than that!"
"The sum of their ages is the number of that house there," she pointed.
"Still... not enough."
"Well, you take your time." She looked at her watch. "I have to go pick up my oldest girl from soccer practice now."
"Your children's ages are..." and the sound of his voice was drowned by a passing tractor-trailer as they both left, leaving me with three people's coffees to pay for!

So, I ask you!
Was that any way to treat a rich uncle?
Also, can you help me figure out what her children's ages are?

[To solve this problem, you will probably need paper and something to write with.]

Odd-numbered Houses Solution
The mathematics professor in the city college lives on the odd-numbered side of a street. The first house is numbered "1" and each house is 2 more than the previous one. There are no gaps in the addresses.

When he comes home at the end of a busy day, he gets off the bus right in front of house #1. Since his mind is full of mathematical conundrums, he finds it hard to remember the number of his own house. Instead, as he walks by each house, he adds its address to the ones before. When the total reaches a certain number that he has memorized, he stops, takes out the keys from his pocket, opens the door and enters his house.

One day, even more absent-minded than usual, he gets off the bus on the high-numbered end of the street. As usual, he adds up house numbers (starting from, as we know, the highest-numbered house on his street). When the total reaches the number that he has memorized, he stops, takes out the keys from his pocket, opens the door and enters his house.

Given that his house number is in two digits, what is it?

[To solve this problem, you could use a tool like Excel. But you cannot use more than three columns or more than 50 rows for your solution.]

Blue and Yellow Hats Solution
In an effort to gauge the perspicacity of their staff, management of a well-known international organization calls a meeting and warns the 38 staff of an upcoming test.

"Tomorrow morning at 10am," the management speaker says, "you will all gather in the corridor on the 11th floor. You will be asked to stand in single file so that each individual can see all the people standing in front of them and none behind.

From a large closed box containing an unknown number of blue and yellow hats, I will place one hat on each person's head.

When I have finished, you will each have one (and only one) turn. Starting at the back of the line, you will say one word – ‘blue’ or ‘yellow’. You cannot use inflection, tonality or other signal -- like a cough -- to impart information in addition to that word. Those who cannot correctly say the color of their hat should leave their identity card in the bin when they leave work at the end of the day. If anyone willingly disobeys the rules, the whole lot of you will be thrown out the window on to the street below where marauding panhandlers will trample you to dust."

So now, the 38 staff have a day to come up with a plan. They have worked well together these past few months and they are loathe to part with their colleagues.

They know that they can all correctly tell blue from yellow -- none of them has a language problem or is colorblind.

What plan does the team come up with to keep the team as intact as possible? What is the greatest risk they manage to limit themselves to?

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