Alien Challenge: Save the Earth!!!!

An energy being from Alpha Centauri comes to your home to test Humanities intelligence.  IF you answer correctly, humanity will become part of a large intergalactic federation. If you answer incorrectly, the alien will shroud the Earth in perpetual darkness.  Here's the problem:

1.  Consider an integer N that is greater than 1.

2.  Consider the integer M, which is the square root of N.

3.  Given that M is the sum the digits of N, what is N and M?

Hint:  Only one value of N can satisfy these properties.

Solution to July 20th's Brain Teasers

First of all the Power of Two Proposal:

I am going to just consider the powers of 2 in order.  2, 4, and 8 do not qualify because they are not multi-digit integers.  16 is disqualifies because 6 is not a power of 2.  32 is disqualified because 3 is not a power of 2. 64 is out because of 6 again.  Now we get to 128.

1 = 2^0
2 = 2^1
8=2^3
And 128 = 2^7.  Obviously this is the smallest, but I wonder what the next one is.

Tom, Dick and Harry Food Venders.

Remember Tom shows up every other day.  Dick every third day.  Harry every 7th day.  They were all together on April 11.  There are many ways to skin this iguana but one can just consider 2,3,7 as multiples and find the Least Common Multiple of all which is 42.  Hence they will all be together 42 days from April 11, which is 23 May.

High School Reunion - 25 years

Last night I attended a special event - the 25th reunion of Ellicott High School's 1988 class.  Sue Smith and I were the only teachers there, which was a shame because there were some very special former students in attendence.

But I get ahead of myself.

1988 - I was 37 years old.  It was my fourth year of teaching in Ellicott, a small town on the plains of Colorado, which was destined to become the model for East Plains, the town in my Bonnie Pinkwater mystery series.  We were in a new building, across the street from the original school, which once housed all thirteen grades of this tiny school district. I was there on the day when the local chapter of the Masons performed their arcane ritual involving oil, salt and wheat on the cornerstone of the new school, which would hold the 7th thru the 12 grades. I knew every teacher and every student in the halls (the 1988 graduating class would have 45 students), including Susan Smith who would become my lifelong best best friend. Together, the two of us would comprise the entire math department.  

In 1984, I followed, Tom McCombe (the new principal and and an old friend) to the school, thinking I would work there for only one year then move on to a bigger school district, one which paid more money, had more programs.  I was broke, newly divorced, had started smoking again.  My first day of work there was a cow standing in my parking place. The original school was run down, 30 miles east of Colorado Springs in what had to be considered the desert.  The populous were ranchers and farmers.  I grew up in Philadelphia population 2 million.  Ellicott had tornadoes and dust storms, for God sake.

By 1988, I couldn't imagine working anywhere else.  The students who would graduate that year, I had watched become young men and women since my first year in 1984.  I had learned to love them, cry with them ( I followed one girl into the bathroom and sat down on the floor next to her as she cried her eyes out. I imagine such behavior would get me arrested today. She would be at the reunion).  I wrote them a song and sang it at graduation - it was my first and wouldn't be my last.  I knew what was happening in the lives of all my coworkers - the problems with their spouses and children, the secrets they entrusted me with, the days when they needed hugs.  Many of them brought their own kids to the school and these would sometimes end up in my classroom. We would commiserate over what was happening at school versus what was happening at home.  Don't get me wrong.  The money was still atrocious. I was always broke, needed a new car, new clothes.  But I loved coming to work.  I stayed for 18 years

Last night I saw these same teenagers again, except they weren't teenagers anymore.  They were older than I was when I was their teacher, most of them now 43.  Many had gray in their hair and beards.  A few were grandparents.  Many no longer resembled the kids in the yearbooks and my memory.  All were a joy to look upon as they told me what they had been up to in the 25 intervening years:  engineers, plumbers, airline attendants, ranchers, general managers of companies large and small.  What's more they remembered me.  We laughed as we recalled things which transpired a quarter of a century ago.  They introduced me to their spouses.  And before the night was over they said the words which were water to my soul.

"You made a difference in my life." 

July 20 Brain Strain: The Powers of Two Proposal

Beezer Wentwistal asked Zenobia Tightbottom (from the famous mathematical family of Tightbottoms) to be his wife.  She answered his proposal with one of her own:

"I will marry you if you can solve a math challenge.  I am thinking of a multi-digit integer N that is a power of 2.  Moreover, each digit in my integer N is a power of 2.  What is N?  If you cannot answer I'm afraid I cannot consider you a viable mate since I require intelligent offspring."

Can you help Beezer marry the love of his life?