think again!

You may use only elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use trigonomery, such as sines and cosines, the law of sines, the law of cosines, etc.

The task is to find the angle x in the diagram, but with your hands tied as explained above. Some people has named the problem: The World’s Hardest Easy Geometry Problem.

I borrowed the diagram from http://thinkzone.wlonk.com/MathFun/Triangle.htm.

- What on earth are you doing?
- Where else?
- ?!
- I am colouring numbers.
- But how is that possible? I thought they were just abstractions, not anything touchable.
- I colour them in my mind.
- And how does it go? I see you use only red and blue.
- I also try to make numbers that differ by 7 or 11 the same colour.
- Isn’t that difficult?
- You tell me! If you can see the colours I am thinking of you shouldn’t have any problems finding that out.

Problem source: Mathcamp 2009.

A building has seven elevators that each can stop in not more than six floors. You can reach any floor from any other floor without changing elevator. How many floors can the building have?

Problem source: The Math Forum.

- Did I tell you about my new neighbour?
- Can’t say that you did.
- Well, he is blind, at least that is what he claims.
- Why do you doubt it?
- You remember the 27 cubes I put together to make a bigger cube?
- How can I forget. I was the one who painted it black on the outside.
- My neighbour accidentally knocked the cube over last night, but when he reassembled the small cubes the big cube was still black on the outside!
- You mean all 54 faces?
- Yes.
- That man can’t be blind!

Problem source: Nick’s Mathematical Puzzles.

In a 3×3 forest there are nine cells. Each cell can occupy a hunter, a tree, or be empty. How many hunters can be in the forest if no hunters can see any other hunter because there is a tree in the way?

The answer is four:

HTH
TTT
HTH

What about 4×4, 5×5, … forests?

Problem creator: Ken Duisenberg.

Find a solution when a, b, and c are all less than 10 and one solution when they are all greater than 10 (or prove that the last one is impossible).

Problem source: Wisconsin Mathematics Science & Engineering Talent Search.

- How much is 23 ha 77?
- What is ha?
- ha is like multiplication or addition.
- In what way?
- If a and b are numbers so is a ha b.
- I see. What more should I know about ha?
- (a ha b) + c = (b + c) ha (a + c) and 0 ha (a + b) = (0 ha a) + (0 ha b).
- And you want to know how much is 23 ha 77?
- That would be nice.

Problem source: Wisconsin Mathematics Science & Engineering Talent Search.

- Give me three distinct primes?
- In what ways distinct?
- Different.
- So you don’t want them to be easily perceived by the senses or intellect?
- Come again?
- Like in the expression ‘it had a distinct flavor.’
- No, I didn’t mean that. Just different would be fine.
- Should they be clearly defined?
- What?
- Like in the expression ‘at a distinct disadvantage.’
- No, thank you.
- To avoid any misunderstanding, did you have in mind that the primes should be very likely or probable?
- ?!
- Like in the expression ‘there is a distinct possibility that she won’t come.’
- I have a feeling that you are avoiding the problem.
- You are flattering me.
- So, will you give me three distinct primes?
- I am working on it.
- Good!
- One final clarification. Should the three primes be notable like in ‘it was a distinct honor and high privilege?’
- I have told you already. They should just be different. Forget distinct!
- When you say different, do you mean dissimilar, separate, various, or unusual?
- Try unusual!
- That’s a hard one. What about 11, 101, and 101111?
- Finally!
- Are they unusual enough?
- Add them and add to the sum their product.
- You mean 11 + 101 + 101111 + 11 * 101 * 101111?
- Exactly.
- I am done.
- Is the answer a prime?
- Any prime or an unusual one?

Problem source: Berkeley Math Circle Monthly Contest.

I am sure you have had the same feeling like I had the other day. I was in a square room 13m and 17m from two of its opposite corners and 20m from a third corner when someone asked me how big the room was.

I was lost for words!

Problem source: Nick’s Mathematical Puzzles.

As everybody knows, time flies. The algorithm below was constructed to find out how fast. How many years, months, and days have passed since, for example, man landed on the moon?

Feed the algorithm 27 January 2010 and 16 July 1969 and it spits out 40y 6m -4d while the correct answer is 40y 6m 11d.

Can you fix it or, much better, make your own algorithm?

FUNCTION AGE

SET DATE TO DMY

M.ageYear=0

M.ageMonth=0

M.ageDays=0

CurrentBirthDate=GOMONTH(_bDate,(YEAR(_chkDate)-YEAR(_bDate))*12)

tmpDate=CurrentBirthDate

M.ageYear=(YEAR(_chkDate)-YEAR(_bDate))

IF CurrentBirthDate>_chkDate

M.ageYear=(YEAR(_chkDate)-YEAR(_bDate))-1

M.ageMonth=0

tmpDate=GOMONTH(CurrentBirthDate,-12)

ENDIF

DO WHILE tmpDate <= _chkdate

IF MONTH(CurrentBirthDate)=2

IF YEAR(_chkdate)=YEAR(CurrentBirthDate) AND MONTH(CurrentBirthDate)=MONTH(_chkdate)

tmpDate=CurrentBirthDate

EXIT

ELSE

M.ageMonth=M.ageMonth+1

tmpDate=GOMONTH(tmpDate,1)

ENDIF

ELSE

M.ageMonth=M.ageMonth+1

tmpDate=GOMONTH(tmpDate,1)

ENDIF

ENDDO

IF _chkDate < tmpDate

IF MONTH(tmpDate)=2 AND day(tmpDate)!=day(_chkDate)

M.ageMonth=M.ageMonth-1

_DateString1=”{^”+allt(STR(YEAR(tmpDate)))+”/”+allt(STR(MONTH(tmpDate)))+”/1}-1″

tmpDate=&_DateString1

ELSE

M.ageMonth=M.ageMonth-1

_DateString1=”{^”+allt(STR(YEAR(tmpDate)))+”/”+allt(STR(MONTH(tmpDate)))+”/1}-1″

tmpDate=GOMONTH(tmpDate,-1)

ENDIF

ENDIF

M.ageDays=_chkDate-tmpDate

RETURN ALLTRIM(STR(m.ageYear))+”y “+ALLTRIM(STR(m.ageMonth))+”m “+ALLTRIM(STR(m.ageDays)) + “d”

The algorithm is written in FoxPro 9 and was found on the Internet. Note that Gomonth() returns the date that is a specified number of months before or after a given date, while Year() returns the year from the specified date.