Philippine Cubers Association

Subject: Re: Exercise your mind. Solve Puzzles! Sat May 17, 2008 2:14 pm

tae pano nya nahulaan!!?!??!?!?!??!?!? TAMA!!!!!!!! Smile

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click

me

Subject: Re: Exercise your mind. Solve Puzzles! Thu May 22, 2008 11:09 am

Old trick na yun. Meron pa nga nung nagfifishing 2 mothers and 2 daughters, they caught 3 fishes but they have one fish each, how could that be? Ayun.

Alfred and Roland have been hired by a farm to sow flower seeds. They've been assigned a 10-acre plot of land and split it in half so they can work independently. Roland starts from the east and Alfred from the west.

Alfred can plow the land at a rate of 20 minutes per acre. Roland takes 40 minutes to plow, but sows seeds at three times the speed Alfred does.

If sowing seeds on the 10-acre plot pays $100, how much of that money should go to Roland

$50 din kasi "They've been assigned a 10-acre plot of land and split it in half so they can work independently"

So kahit gano sila kabilis, pag natapos na ng isa sa kanila yung half, hanggang dun nalang, hindi sila pwedeng mangagaw ng job. Haha, tama po ba?

4x4x4 Age : 18 Joined : 24 Oct 2007 Posts : 391

hey, the question is "HOW MANY?".. not "enumerate".. well anyway, the final answer must be a NUMBER.. haha..but he also said we should include a proof. I would like to solve this not just by enumeration, but by set definition. hehe. RECALL YOUR HIGH SCHOOL ALGEBRA (SETS in particular) FOR MY SOLUTION.. HAHA

My set of set definitions will be based on the case only where there are EXACTLY 3 CONSECUTIVE same digits from the rightmost side. Although elegantly, makukuha ko rin ang times na may 4 consecutive digits:
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i define the set A={0X:XX | X is an element of {0,1,2,3,4,5}}

>>>since minutes are only up to 59, meaning 55 will be the last minute with same digit used. this means |A|=6>>> six ang cardinality ng set na ito, obvious naman right? This set also includes 00:00 which is one of the times na may 4 consecutive,same digits

I also define the set B={1X:XX | X is an element of {0,1,2}}

>>> WHY only {0,1,2}? kasi, wala namang 13:33 na oras diba? hanggang 12 lang ang hours sa 12-hour scheme. Also, included na dito ang isa pang time na may 4 consec. same digits used which is 11:11. |B|=3.
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So here the next set of definitions that is defined for the times that there are 3 leftmost, consecutive, same digits

C={XX:XY | X is an element of {0,1}, Y is an element of {0,1,2,3,4,5,6,7,8,9}}

C, implicitly said na but as an emphasis, is SUBSET OF X x Y (x is read as "cross")

|C|=|{0,1} x {0,1,2,3,4,5,6,7,8,9}|=|{0,1}| times |{0,1,2,3,4,5,6,7,8,9}|= 2 times 10 = 20

BUT AGAIN, this definition has included our special cases, namely 11:11 and 00:00. to prevent the repetition of counts later, it is important to get |INTERSECTION OF A and C|=1 and |INTERSECTION OF B and C|=1

\\\note that |INTERSECTION OF A and B|=0, so we dont really need it.\\\

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MALAPIT NA MATAPOS GUYS!!!
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RECALL the CARDINALITY OF UNION OF (3) SETS FORMULA:

|A U B U C|=|A|+|B|+|C|-|INTERSECTION OF A and C|-|INTERSECTION OF B and C|-|INTERSECTION OF A and B|

=6+3+20-1-1-0=27 ang times na may "three or more of the same number in a row" !!!!

EDIT:wait.. i forgot something, 00:AB pala ay wala sa digital clock, 12:AB pala yun.. so we minus the set
D={00:XX| X is an element of {0,1,2,3,4,5}}

|D|=6

SO yung 27-6=21

BUT ang sabi, "over the course of one day".. so ilang beses ba natatapos ang pagbibiliang ng digital clock na may 12-hr scheme in a day? simple arithmetic, 24/12=2... so finally, (2)(21)=42..

54 ANG sagot ko.

EDIT:MALI PALA ANG SAGOT KO... haha
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LET:
>>cxcxc be a Rubik's Cube
>>U be the set of all tangible Rubik's Cubes,
>>i.e., U={cxcxc | c is an integer and lies in the interval [2,11]}
>>C be the set of my cubes on hand
>>S={2, 3, 4, 5}
MY CUBES ON HAND CAN BE THE DESCRIBED BY THE SET C where...
>>C={nxnxn element of set U | n is an element of S}

UPD. MATH MAJOR. PERIOD.