need the angles at each end of the cutting line to sum to pi, so they’ll

have to be the same angle twice. ]]>

Suppose a > b. Call the largest number of each row “big” and the second largest number of each row “small”. (big + small = a). Each column contains only one of the “big” numbers of each row since each “big” > a/2.

Now find the smallest of the “big” numbers (say B). In that row, look at the “small” (2nd largest number in the row, say S). In the column containing this S, there is a “big” number, say BB. BB + S >= B + S since B is the smallest of the “bigs”. But B + S = a > b. So BB + S > b. Contradiction. A similar argument works for b > a.

]]>9 years

example: 1899 1908 and party in 1926 (ages 27, 18) ]]>

The square has c rows and c collumns.

The sum of all numbers equals the sum of all numbers horizontally alligned and also equals the sum of all numbers vertically alligned. We will call it d.

d / c = a and d / c = b, therefore a=b ]]>

– AD have been switched, then ACD = 8 and BF = 8, both sides weigh the same

– BE have been switched, then ACD = 8 and BF = 11, so right side is heavier

– CF have been switched, then ACD = 11 and BF = 5, so left side is lighter.

Since all three give different outcomes, the side of the balance (or if it doesn’t move) tells which balls have been switched. ]]>

Problem 2: Let Paul’s age = 19ab where a,b have possible values 0,…9, then possible age range for Paul is 1+9+0+0=10 …. to 1+9+9+9 = 28. Similarly Peter’s age ranges from 9 to 27. But peter is older than paul therefore Peter’s age range narrows to 11 —>27

and Paul’s from 9 —>26. Earliest year in the 20th century when Paul could be born is 1900 which would make the year they are celebrating their b’days 1910, when Paul would be 10 (1+9+0+0 = 10 and 1910-1900 = 10) latest year is 1926 which would make peter 27 if born in 1899. By examiming possible combinations, answer is that age difference is 13 years. (not very elegant but best i coudl do)

Peter could be born 1910-27 = 1873 –> 1899 (1910-1899=11) 1+8+7+3 = 19, 1+8+7+4 = 20,

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