Home	Solutions for Professor Tangent's Brainteasers

We saw in the "hints" section that 3 balls require 2 balances to get a final answer. Since 12 balls contain 4 groups of 3, and it takes 2 balances to determine which group of 3 contains the different ball, the final answer would require 4 balances. This suggests that we need to deal with groups of 4 or more to have a chance.

Begin by balancing 4 and 4. If they balance, one of the 4 remaining balls is different. Now choose 3 of the remaining 4 to balance against any 3 of the known good balls. If they don't balance, you've at least determined whether you're looking for a lighter or heavier ball, and we saw previously that it only takes one balance to determine which of the 3 is the culprit.

If the first 2 groups of 4 don't balance, it's a bit trickier. Let's suppose the left side is heavier, but remember that there could be a lighter ball on the right side. For the 2nd balance, replace 3 balls on the left (heavy) side with 3 balls from the remaining 4, and in addition, swap the 4th ball on the left side with any ball from the right side. If the scale now balances, you know that one of the 3 balls removed from the left side is heavier. If the left side is now lighter, one the 2 balls swapped is different. If the right side is still lighter, you know that one of the 3 balls on the right side that wasn't swapped is lighter.

An orange because motorcycles don't have doors.

There's no such thing as half a hole!

Just observing which lights are on or off after one trial doesn't determine more than one switch/light pair with absolute certainty. Lights also produce heat, and get hotter over time. Turn on one light, wait 10 minutes, then turn on another. Quickly check the temperature of the lights.

The "easy" clues should get you this far:

	House #1	House #2	House #3	House #4	House #5
Color		blue
Drink			milk
Vehicle
Cigar
Job	physicist

The green house is next to the white house, so the green house is either #3 or #4. It can't be #3 since the milk drinker lives in #3 and the coffee drinker lives in the green house. The only color left is yellow, which allows filling in a few more. Now you should be this far:

	House #1	House #2	House #3	House #4	House #5
Color	yellow	blue	red	green	white
Drink			milk	coffee
Vehicle		Audi
Cigar	Dunhill
Job	physicist		manager

We know the beer drinker smokes Bluemasters, and the buyer drinks tea. These pairs can be in either #2 or #5. Either way, the only beverage left is water, which is drunk by the physicist in #1, who also has a neighbor smoking blends. The Bluemaster smoker therefore lives in #5.

	House #1	House #2	House #3	House #4	House #5
Color	yellow	blue	red	green	white
Drink	water	tea	milk	coffee	beer
Vehicle	minivan	Audi	Buick	Harley	Volvo
Cigar	Dunhill	Blends	Pall Mall	Prince	Bluemasters
Job	physicist	buyer	manager	engineer	designer

For the sum to contain 5 digits, the sum of S+M must produce a carry and M can only represent 1. S can only be be 8 or 9 depending on whether the sum of E+O produced a carry. Since M must be 1, the sum of M+S+carry is either 10 or 11. O cannot be 1 since M already represents 1, so O must represent 0. The sum of E+0 cannot produce a carry (sum would be 10) or N would also represent 0. For the sum of S+1 to be 10 (with no carry from E+0), S must be 9. So far we have:

9END

+10RE

10NEY

The sum of E+0 cannot be E, so the sum of N+R must have produced a carry. So far we know that:

1. N = E+1(carry from N+R)
2. N + R = E or N + R + 1(carry from D+E) = E

Replacing the "N" in equation 2 with "E+1" (from equation 1) we get: E+1+R=E or E+1+R+1=E

The only thing we can add to E to get a sum of E is 10. R must therefore either be 8 or 9. Since S already represents 9, R must be 8 and the sum of D+E must produce a carry. Now we have:

9END

+108E

10NEY

Since Y cannot be 0 or 1 (they're already represented by M and O), the sum of D+E must be at least 12 to produce a carry. Since 8 and 9 are already taken, D+E must represent either 7+6 or 7+5. E cannot be 6 or N would be 7 and either D or E must represent 7. E must represent either 7 or 5. If E represents 7, then the sum of N+8+carry would be 17 which requires that N represent 8. R already represents 8, so E must represent 5 and D must represent 7.

95N7

+1085

10N5Y

The other letters are now easily converted to:

9567

+1085

10652

Adding letters to a 5-letter word will never result in a word with fewer than 5 letters. We therefore must consider other possible interpretations for "becomes shorter". We could punctuate the sentence to read "What five-letter word becomes 'shorter' when you add two letters to it?", meaning what word becomes the word "shorter"? The answer is the word "short".

The 4-cup container can measure exactly 8 cups. We need 1 more cup. The difference between 7 and 8 is 1. If you pour 4 cups into the 7-cup container, refill the 4-cup container and fill the 7-cup container from the 4-cup container, there will be 1 cup left in the 4-cup container.

A total of $30 and each man having $10 suggests a total of three individuals. Father/son status isn't as certain, as one man may be both at once. A son/grandson, father/son and father/grandfather solves the apparent dilemma quite nicely.

Since there were only 2 red hats in the bag, if Rudy saw red hats on Trudy and Judy, he would have known is own hat was green. Therefore Rudy saw at least 1 green hat on either Trudy or Judy. If Trudy saw a red hat on Judy, she would have known that the green hat Rudy saw was her own. Rudy's and Trudy's silence allowed Judy to deduce that her own hat's color must be green.

One question that would be answered differently by T-towners and F-towners if they both told the truth is "which road will you take to get home?" Either way, the answer will indicate the road to T-town. To exploit the truth about a lie, or a lie about the truth principle, one possible question would be "if I asked someone from the other town which road leads to T-town, what would the answer be?". If the person you ask is from T-town, the neighbor from F-town would falsely indicate the road to F-town, and the T-towner would tell the truth about this lie. If the person you ask is from F-town, the neighbor from T-town would truthfully indicate the road to T-town, and the F-towner would lie and indicate the road to F-town.

At the market, Moe essentially calls himself a liar, so his comment is false. Therefore Miney and Eeny speak the truth while Meeny lies.

If suspect #3 speaks the truth, then #1 and #2 are both liars and guilty which isn't possible. #3 therefore lies which means that #2 is innocent and truthful. #1 is therefore a liar and guilty. Another possible analysis is to observe that #1 and #2 cannot both be liars or both would be guilty. Therefore either #1 or #2 is innocent and truthful. #3 therefore lies.

This is really just a simple calculation once you understand the details. Since "cold" is really just absence of heat, it's simpler to think of losing half of whatever heat is there to start with. Two familiar temperature scales are Fahrenheit and centigrade. Either way, at 0 degrees, there's still heat left, so we could take half of it away. The Kelvin temperature scale is different. 0 degrees Kelvin is also known as "absolute zero", which means that all heat is gone. Absolute zero is about -273 degrees centigrade, or about -459 degrees Fahrenheit. To find "half" of any temperature, convert to the Kelvin temperature scale and take half of that number.

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