Math and Logic Puzzles Collection

Area is proportional to the square of the radius. If the radius becomes 0.9r, the area becomes (0.9)^2 = 0.81 of the original area. The decrease is 1 - 0.81 = 0.19, or 19%.

Upstream speed = 20/6 = 10/3 kmph. Downstream speed = 18/4 = 4.5 kmph. Speed in still water = (4.5 + 3.33)/2 = 3.91 kmph. Speed of current = (4.5 - 3.33)/2 = 0.583 (7/12) kmph.

At 4:00, the hour hand is at 120° and the minute hand is at 0°. The minute hand moves at 6°/minute while the hour hand moves at 0.5°/minute, so the minute hand gains 5.5° per minute on the hour hand. To align, it must gain 120°. Time = 120/5.5 = 21.8181 minutes = 21 minutes 49.09 seconds. Option A (4:21:49.5) is the closest approximation.

By the pigeonhole principle, with only 2 color options (black and brown), picking 3 gloves guarantees at least one matching pair. The worst case is picking one black and one brown first, but the third glove must match one of those colors.

The shadow length is directly proportional to the pillar height when the sun's angle is constant. Since the height doubles from 8 to 16 feet, the shadow also doubles from 4 to 8 feet.

For the number 54, the sum of the digits is 5 + 4 = 9. Multiplying the sum by 6 gives 9 * 6 = 54. No other option satisfies this condition.

The number of handshakes in a group of n people is calculated by the formula n(n-1)/2. For 10 children, it is 10 * 9 / 2 = 45.

5 painters take 8 days, so the total work is 40 man-days. After 4 days, 5 painters have done 20 man-days. 20 man-days remain. With 10 painters, it takes 20/10 = 2 more days.

The ratio of the pillar to its shadow is 6:2, which simplifies to 3:1. If a pillar is 15 feet tall, its shadow will be 15 / 3 = 5 feet long.

Average speed is NOT the arithmetic mean of speeds - it's total distance divided by total time. Distance = 55 miles each way = 110 miles total. Time uphill = 55/28 hours, time downhill = 55/70 hours. Total time = 55(1/28 + 1/70) = 55(98/1960) = 5390/1960 hours. Average speed = 110 / (5390/1960) = 110 × 1960/5390 = 40 mph. The arithmetic mean (49 mph) would be wrong because you spend more time at the slower speed.

By the Pigeonhole Principle, if you have two colors of socks, taking three socks guarantees that at least two must be the same color (either 2 white/1 black or 2 black/1 white).

This is a 'look-and-say' sequence. Each term describes the digits of the previous term. The term '13112221' is described as: one 1, one 3, two 1s, three 2s, one 1, resulting in 1113213211.

Let daughters = d, sons = s. Each daughter has d-1 sisters and s brothers, so d-1 = s. Each son has d sisters and s-1 brothers, so d = 2(s-1). Substituting s = d-1: d = 2(d-2), giving d = 4 and s = 3. This works because each daughter has 3 sisters and 3 brothers (equal), while each son has 4 sisters and 2 brothers (twice as many).

Let the cork be x. The bottle is x + 100 cents. Total: x + (x + 100) = 105 cents. 2x = 5, so x = 2.5 cents. This is a classic cognitive reflection test question.

Take one chain (3 links) and break all three links. Use these three open links to join the remaining three chains together into a single loop. This requires only 3 cuts and 3 welds.

According to Archimedes' principle, ice floats because it displaces a volume of water equal to its mass. When ice melts, it becomes exactly that amount of water, so the total volume remains unchanged and the water level stays the same. This is true regardless of the ice cube's shape or size, as long as it's floating.

The pattern is: First Number * (First Number + Second Number). So, 2*(2+3)=10; 7*(7+2)=63; 6*(6+5)=66; 8*(8+4)=96. For 9 and 7: 9*(9+7) = 9*16 = 144.

As the tide rises, the boat floats higher on the water. Because the ladder is attached to the boat, it rises with the tide, meaning the water level relative to the rungs never changes.

If there are 2 or more honest politicians, choosing those 2 would give 0 corrupt, violating the condition. With exactly 1 honest, any pair including that person plus any corrupt politician satisfies 'at least one corrupt.' Thus, 99 corrupt and 1 honest is the only solution.

The sequence represents the last and first letters of consecutive months: SeptembeR/OctobeR (SO), NovembeR/DecembeR (ND), Januay/Februay (JF), March/April (MA). The next are May/June (MJ) and July/August (JA).