Solution To The Puzzle For The Moderator

As the title of the puzzle suggests, the challenge was directed on me. But I never had the chance to try if I could handle it or not. Most probably not. Fortunately, two former students of mine stood up and save me from this puzzle quandary. Engr. Jeffrey Agbunag and Engr. Ramil Balisnomo belong to the last batch (and therefore youngest) of the CE students I handled for Reinforced Concrete Design. They now work in Manila as Structural Design Engineers and are planning to join us in Dubai in the near future.

For further clarity, I made minor alterations to their solution and added some explanations without changing the essence of their reasoning. So here it is.

A Puzzle For The Moderator

There are three positive integers, each with two non-repeating-digits. The sum of the first and the second integers when added to thrice the third integer results to 242. When half of the second integer is subtracted from a number which is the reverse of the first integer, and the difference is added to a number which is the reverse of the third integer, we obtain 116. If we reverse the second integer and deduct the resulting number from the first integer, we get a difference of 10.

Amusingly, the third integer has other amazing properties. When the number and its reverse is divided by two, the halves are reverses of each other. The halves when further halved are also reverses. Can you give me the integers?

The Solution:

by Jeffrey Agbunag and Ramil Balisnomo
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Let :
A = first digit of the first integer
B = second digit of the first integer
C = first digit of the second integer
D = second digit of the second integer
E = first digit of the third integer
F = second digit of the third integer
Then:
10A + B = first integer; 10B + A = the reverse of the first integer
10C + D = second integer; 10D + C = the reverse of the second integer
10E + F = third integer; 10F + E = the reverse of the third integer

From Condition 1:
(10A+B)+(10C +D) + 3(10E+F) = 242

From Condition 2:
(10B + A) – 0.50(10C + D) + 10F + E = 116

From Condition 3:
(10A + B) – (10D + C) = 10

The three equations when simplified and written in matrix form becomes:

The non-square coefficient matrix will show that no unique solution is possible because there are only three equations and there are six unknowns. We would therefore require three more independent equations. In order to produce the extra conditions, the duo came up with two remarkably original propositions:

The second paragraph of the puzzle talks of the third integer and its reverse having their halves and quarters as reverses. Thus,

1. The Agbunag Theorem:

“For the set of positive, single digit numbers, two - and only two - could be halved twice and still produce whole number results.” These are the numbers 4 and 8. Thus, they concluded that only the two-digit numbers 48 and 84 could satisfy the conditions of the second paragraph. However, they correctly pointed out that it could not be 84 because it would contravene the stipulations of Condition 1 (the sum would exceed 242 since 3 x 84 is already 252). Hence, the third digit is 48 and therefore

E = 4 and F = 8.

Adding these two equations to the matrix expression, we get

Still, there are more unknowns (6) than equations (5). So from where could we possibly acquire the 6th equation?

Enter Jeffrey’s and Ramil’s numerical acuity; and they come up with a simple but universal truth where most of us could only see a blank wall.

2. Balisnomo’s Theorem:

“If the difference between two 2-digit numbers is 10, their last digits must be the same.”

Thus, hidden in Condition 3, they pointed out that B must be equal to C by Balisnomo’s Theorem. In equation form:

B – C = 0

which when inserted into the matrix expression, we finally obtain an invertible coefficient matrix.

Explicitly, we obtain and write the inverse and perform the matrix multiplication, using Excel:

to obtain:

finally,

1st integer = 36
2nd integer = 62
3rd integer = 48

sana tama sagot namin. Hehehe.

God bless.

-Jeffrey and Ramil
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I have confirmed this with Carlo. TAMA ang sagot niyo! Congratulations and my admiration to both of you.

- icarus.
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Solutions for the other Problems:

A certain Marlon solve two out of the three problems which I posted as additions to Carlos lone puzzle. I am not sure if Marlon is a Louisian but he is one heck of a logical guy. I hope he will come forward and identify himself because he is surely a welcome addition to our rank. Here are the postings he made on the comment sections:

Marlon said...

Given : ABCDEx 4 = EDCBA
The answer is : 21978x 4 = 87912

But rather than just give you the answer, here's how I figured it out.

First, it is obvious that A must be an even number, because we are multiplying by 4 (an even number). The last digit will therefore be even. It can't be 0, because that would make ABCDE a four-digit number. It can't be more than 2, because that would result in a six-digit answer.

So A = 2.

2BCDEx 4= EDCB2

So what can E be? The choices are E = {3, 8} because 3 x 4 = 12 and 8 x 4 = 32. But a value of 3 doesn't work in the result (3????) because it is too small.

2BCD8 x 4= 8DCB2

Since the final number is 8 and we have 2 x 4, that means there is no carry from the prior multiplication (4 x B + carry). So B can't be anything higher than 1, possibly 0.Looking at the other side of the equation, we have 4D + 3 = (a number ending in 0 or 1). In other words, 4D must end in 7 or 8.

Obviously only 8 works, because 4 is an even number. Working forward again, that means B = 1

21CD8 x 4 = 8DC12

So what values of 4D result in a number ending 8? 4 x 2 = 8, 4 x 7 = 28. Now 2 is already taken and the problem said the digits were unique. So D = 7.

21C78 x 4= 87C12

Finally, we have a carry of 3 (from 28 + 3 = 31). And when we calculate 4C + 3 it must also result in a carry of 3 and a last digit of C.

In other words: 4C + 3 = 30 + C

This is easy to solve:

3C = 27 ; C = 9

Thus the final answer is: 21978 x 4 = 87912

February 19, 2009 9:10 PM
Marlon said...

The secret is to notice that the answer has more letters (5 letters) than the question (4 letters).

That M at the beginning of money is a carry from the thousands place, so M = 1.

Now we have:

SEND+ 1ORE = 1ONEY

Now, in the thousands place there is a 1, so the only value for S that could cause a carry is S = 9 and that means O = 10.

Now we have: 9END+ 10RE = 10NEY

Now look at the hundreds place. If there were no carry from the tens place, E and N would be the same because E + 0 = N, but E and N can't be the same, so there must be a carry from the tens place.

Now we have: 1 1 <-- carry9END+ 10RE = 10NEY

Now the equation for the hundreds place is 1+E+0 = N or just 1+E = N. In the tens place we can have N+R = E+10 if there is no carry from the ones place, or we can have 1+N+R = E+10 if there is.

First test: no carry from the ones place:N+R = E+10 and 1+E = N(1+E)+R = E+101+R = 10R = 10-1R = 9

But S = 9, so R cannot = 9.

That means there is a carry from the ones place and we get:

1+N+R = E+10 and 1+E = N1+(1+E)+R = E+102+R = 10R = 10-2R = 8

So now we have: 1 11 <-- carry9END+ 108E = 10NEY

N cannot be 0 or 1 because 0 and 1 are taken. N cannot be 2 because 1+2+8 = 11 and then E would equal 1, but it cannot equal 1 because 1 is taken.

N could equal 3,4,5,6 or 7 but it cannot equal 8 or 9 because 8 and 9 are taken (and E must be 2,3,4,5 or 6 because it is 1 smaller than N).

If E were 2, then for the ones place to carry D would have to be 8 or 9, and both 8 and 9 are taken, so E cannot be 2 (and N cannot be 3).

If E were 3, then for the ones place to carry D would have to be 7,8, or 9, but D cannot be 7 because then Y would be 0, which is taken, so E cannot be 3 (and N cannot be 4).

If E were 4, then for the ones place to carry D would have to be 6,7,8, or 9. D cannot be 6 because Y would be 0, but D cannot be 7 because Y would be 1, and 0 and 1 are both taken, so E cannot be 4 (and N cannot be 5).

The only two possibilities for E now are 5 and 6. If E were 6, then N would be 7 and D would have to be 4 (which would make Y = 0), 5 (which would make Y = 1), 6 (which is taken by E), or 7 (which is taken by N).

There are no solutions for E = 6, so E must be 5.

So now we have: 1 11 <-- carry956D+ 1085 = 1065Y

Doing the same reasoning for D and Y and get the answer

9567+ 1085 = 10652

February 19, 2009 9:21 PM
ICARUS said...

Crisp and precise logic, nothing wasted, each line, each thrust and stroke goes straight for the jugular. Well done, Marlon!

I can tell you're a puzzle enthusiast yourself. Maybe you can send us some more interesting posers.

Thanks a lot.
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Amusing Challenge III

Carlo sent in an amusing challenge problem and calls it “a puzzle for the moderator”. Obviously, he’s got enough of my brain-teasers and this time he wants to be the one to bewilder; and not the bewildered. It’s the young lion of the pride, newly emerged from being a cub, playfully testing his power against the aging alpha male. It’s the promising pupil gauging his washed-up teacher, the growing son measuring his fading father. And this is all very healthy. As in everything, the new will shade the old - in due time. I am, of course, much obliged to solve it but before I do, I will give everyone the chance to test their mettle against it. So here it goes…

A Puzzle For The Moderator
by Engr. Carlo E. Chan

There are three positive integers, each with two non-repeating-digits. The sum of the first and the second integers when added to thrice the third integer results to 242. When half of the second integer is subtracted from a number which is the reverse of the first integer, and the difference is added to a number which is the reverse of the third integer, we obtain 116. If we reverse the second integer and deduct the resulting number from the first integer, we get a difference of 10.
Amusingly, the third integer has other amazing properties. When the number and its reverse is divided by two, the halves are reverses of each other. The halves when further halved are also reverses. Can you give me the integers?

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And for your further amusement, here are three other challenging number puzzles in addition to Carlo's.

Each letter in the three problems that follow represents a unique numerical digit (for that particular problem). Figure out the number digits represented by each letter so that no inconsistency or repetition arises when the indicated operations are performed.

As usual, send your solutions with the accompanying explanations on how the answers were arrived at.



Happy weekend, everyone!




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The Summer of 93: A Short Story

by Rowell Olivar

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This story started when I was on my 3rd year in CE, in good old Saint Louis.

Things had been very tight financially at home. Thanks to Mrs Laureta I still had my CICM scholarship(though my grades were predictably bad), and Yit was to start teaching at the high school. But, with two sisters in college, and another two in high school, our day to day existence was a struggle.

The school year has just finished so I was asking around for any summer jobs that might be available. Mrs Laureta suggested I try for the school canteen. The lease has expired and the management has been turned over to the college. As luck would have it, the new manager, Mrs Cadiente took me on as waiter/helper for the summer.

There were 5 student helpers. Me, Edmond Corpus, Gina Nicol, Hazel Jubilo, and a young girl from Luna (I can’t remember your name sorry nevermind). We also had our cook, Edgar from Bauang, Akong (my future Ninang Arsen’s nephew), Ate Rogie, and Mrs. Cadiente, the lady manager. There was also my good friend Bert, the accountant, and Edwin, his Beatlemaniac assistant.

We boys had a nice racket. It seemed that the guy delivering Zest -O tetrapacks wanted the empty tetras back, and was willing to pay for them too. We were very glad to accommodate, of course. An empty will fetch 50 cents back then. In one day, our high school clientele alone will be around 100, and the college kids about the same. Every weekend the take will be split evenly, with some being spent ‘for our girls’ Hazel, Gina and the rest. It was my first taste of ‘engineering economy’, even before Engr Rubie Delizo Nones failed my buddy Micael the subject, haha. But that’s another story.

Those were exciting times. I washed dishes, cleaned toilets, and waited on tables. Besides that, I had the time of my life flirting with Hazel and Gina, no offense to their husbands today.

When the manager and Ate Rogie weren’t looking, the cook and Akong used to set aside the choice cuts of fish and meat, and prepare meals especially for us and the girls. God bless Edgar, his ‘Dinakdakan’ and ‘fish Kilawin’ are the best I have come across.

With the passage of time, I guess it won’t do any harm to say that we used to look at the lady teachers and give them scores. Ladies, to tell the truth, back then the ‘Dean’s List’ in that department used to include the Laron sisters, Engr. Collado (now Doctolero), and a new one we used to call Anjanette Abayari, Ms. Taguba. Bert wanted to include Mang Felix the cleaner and Dacanay the security guy, but he was voted down, to his never ending objections.

It didn’t occur to me, but I must have smelled terribly. Our work involved a lot of sweat, and like a poor boy, I didn’t have any idea what a deodorant was. Sometimes ‘sosi’ students from Manila, mostly making up for failed subjects, would show up. These are the type to avoid. They are neither Ilokana nor Tagalog. They speak to the waiter in a queer sort, like Kris Aquino used to do before she found Ipe:

“ Puede make tapon this wrapper, hinde sia biodegradable eh. Alam mo what biodegradable is ano?”
Or,

“My goodness, why don’t you gamet rexona ha? Ansama ng smell mo!”

Puweh! I smelled bad, egskyus mee, medem, but pleese kiss my puwet. har har

It wasn’t like that often. There was a lot of good people in the world.

Mang Leon , for instance. He was an all around guy for the college then. He showed me how to use the hand trolley for moving cases of softdrinks around. Whew! It wasn’t easy to push 5 cases, piled one on top of the other, from the canteen at the back of the chapel to the gym.
Mang Mundo was also very helpful when it came to wiring and plumbing. He would appear near the counter and I would yell ‘Pasan ko ang Mondoooo!’ It never failed to make Hazel giggle.

That summer of 1993, there was a promo by coca cola. In every case, there were 5 free bottles. I spent most of my time peeping under the caps and setting these aside for future use. Bert, my loving mentor, had this great idea of giving freebies to deserving ladies. No free bottles to gentlemen, if you please.

Bert had a friend, Eric Guron, who was then a staff with Nancy Lopez (Attorney Lopez-Bilaoen today) working on their yearbook. It was from this guy I first heard that hideous song ‘Bizarre Love Triangle’.

Eric sometimes brought along Nancy, Jaime Collado, David Bilaoen, and two girls Mary Grace Canono and Daisy Sayangda for snacks. I knew these beings by sight. During the regular year they are staffs at the Torch with my brother Yit. Nancy was Yit’s academic rival, Grace was his oh-so-secret crush, Daisy was the daughter of a town mayor, and Jaime and Dave were the girls’ bodyguards.

To be fair to Eric and the boys, they are good-looking in a manly, boyish, manly, boyish, sense. But you can’t blame us canteen guys if our interest is on the three ladies. Nancy is voluptuous, was an editor of the Torch and the yearbook, Student Council president, Magna Cum Laude, surely destined for great things. Grace is slim, tall, elegant, definitely a class above most. And Daisy is a true Filipina beauty- slender, morena, long legs and straight black hair, plus an all around scholar. Nancy has just graduated and is shortly to be married to David. The other girls will be entering their senior year the coming June.

I do not recall if Edmond or Akong ever served snacks to them. Whatever, I looked forward to giving these girls their bottles of soda, and loving the look on their faces when they found their drinks were free.

Now, 16 years later, I see how stupid I must have appeared- trying my best to impress using free cokes. Daisy took notice, however, and was very nice. I would hang around the table next to theirs and pretend to wipe off dust or crumbs, always on the lookout if the door to the manager’s office would open. She included me in their talk. Ever the opportunist, my two cents’ worth was always forthcoming. She was so nice she laughed at my jokes.

That summer was special.

For one, it showed me that a person doesn’t need to be rich to get along. You can show people that, though poor, you are serious in trying to better your lot. A little sweat, a little hard work, and you will see- help will arrive in so many forms.

When the regular year started, Daisy and me became close friends. It seemed inevitable that someday we will fall for each other. But ask Dr. Dag-o, and she will swear it took one year before that happened. Why? I needed that much time to convince her that she is a wee bit smaller than I am.

Happy Valentine to my wife, Daisy.

Here Comes the Rain

Her name is Sue but she likes to be called Rain. She’s a Louisian through and through, earning her degree from SLU, Baguio. I met her in cyberspace through the Asingan Journal which is a very interesting blogsite about our hometown by our common friend, Vic Costes. It was there where I first read and got hook with her works. Rain has that certain lyrical sensitivity and romantic awareness to see poetry in even the simplest things around her. She writes as she feels and thus her verses are free-flowing and spontaneous; And always from the heart.

I never got to ask her about her self-chosen name. It's maybe because rain brings in freshness and quenches our otherwise parched part of the earth. Or, It’s maybe because of the rainbows in the aftermath. In that sense, the name fits her to a T…


Love
by Susan Villanueva

Love has no season,
It has no clime…
It transcends smoothly
with the passage of time...

It touches us
from birth to our death.
It is with us in each gasps,
in each and every breath...

Our lives are sustained
by our efforts to Love…
Love holds us up
to the end of the day

when the light
fades away…

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A Sonnet On Undying Love

This month, February, is universally associated with matters of the heart. Maybe it’s because of the cold balmy weather, quite conducive to courtship and romance. Or maybe because of Valentine’s Day falling on the 14th. Whatever it is, people – Engineers included and perhaps more so - are a little more passionate and a little more in love on this month.

Quite fittingly, we start the month with a Love Poem from one of my friends from Asingan, Pangasinan.

Sammy was my softspoken school mate from the Asingan North Central School. He was a year ahead of us but we were closely acquainted through Scouting. But even before that, I look up to Sammy because his passion and talent for the visual arts is something out of the ordinary. In those Art Corners of every classroom then were posted the outstanding artworks done by previous pupils who are already in the next grade. These are posted for us in the lower grade as samples to emulate. Seven out of ten of those works, maybe more, are by Sammy Antonio. How could you not idolize such a guy?

We parted ways in High School as we went to different institutions. Much later, I learned he was teaching design and drawing subjects at The Pangasinan State University. Then we lost contact completely. One day, I found him again in the internet. It turned out that he migrated to the States to fulfill his passion for arts to become the Exhibit Designer And Preparator, International Museum Of Arts And Sciences, Mcallen Texas. Definitely bigtime but still innately humble, Sammy has not forgotten his Asingan roots and remains charitable with, among others, his scholarship foundation for poor but deserving PSU students.

In the sonnet that follows, Sammy raptures about his Chrysanthemum, a lovely Nurse from Asingan who worked in Texas and whom he pursued and regaled relentlessly with his fine paintings and love-laced poetry. They eventually got married and were blessed with two charming sons. As fate would have it, she passed away from cancer some six years ago. But she lives on in Sammy’s immortal poetry...
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MY EVERLASTING CHRYSANTHEMUM

by sammy d. antonio

Let me refresh the foliage of thy mute yesteryears,
You were then a blooming flower I was young bee frail;
Roaming around your garden hoping that you may hear,
The whisper in my heart i cant even say it clear.

I felt sad when you went away, bound for a new place;
In my lonely world, i gazed at yellow daffodils.
Yet in my dreams my dear Chrysanthemum its your face,
Because of your everlasting qualities which the other flower fails.

Now that i found you, i can no longer conceal;
The whisper in my heart, it's a must i will now reveal;
I will even shout for the whole world to hear,
"I love you Chrysanthemum", it's no longer a whisper;

You may call me ambitious, a dreamer reaching you above;
Yes, because i have nothing to offer but my humble love.
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