Math and Logic Puzzles

A Harshad number is an integer divisible by the sum of its digits. For 800, the sum is 8+0+0=8. Since 800 is divisible by 8, it is a Harshad number. 802 (sum 10), 805 (sum 13), and 809 (sum 17) are not.

A circle has infinite diameters because any line passing through the center to any point on the circumference is a diameter. Since there are infinite points on a circle's circumference, there are infinite possible diameters. This is a fundamental property of circles.

Percentage increase is (New-Old)/Old. From 13 to 81 is ~523%. From 12 to 80 is ~566%. From 1 to 69 is (68/1)*100 = 6800%. Thus, 1 to 69 is the greatest increase.

Surface area of a cube is 6y^2. Given 6y^2 = 294, then y^2 = 49. Taking the square root gives y = 7.00 cm.

A regular heptagon has 7 equal sides. If its perimeter is 84 cm, each side measures 12 cm. However, the question only states that 'a length of a side' of the square equals 'a length of a side' of the heptagon - this is grammatically ambiguous. It could mean any one side length, not necessarily all sides being equal. Without clarity on whether the square's side equals the heptagon's side length (12 cm), the area cannot be definitively determined.

The negative square root of 36 is -6. Thus, t = -6. Substituting into y = 2t + 28 gives y = 2(-6) + 28 = 16. Area = (x+y)(x-y) = x^2 - y^2. Substituting y=16 gives x^2 - 256.

The stick is 1.12 meters (112 cm) long. The total ratio parts are 1+3+4 = 8. Ben's share is 4/8 (or 1/2) of the stick. 112 cm / 2 = 56 cm.

Using the exponent rule 2^(x + y) = 2^x * 2^y, we substitute the given values: 7 * 38 = 266. The other options are incorrect applications of algebraic rules.

Daily production = 10,000 litres = 10,000,000 ml. February 2000 was a leap year (2000 divisible by 400), so it had 29 days. Total = 10,000,000 x 29 = 290,000,000 ml. The conversion is litres to milliliters (multiply by 1000), then multiply by days in February.

The numbers are -8, (2 - [-8]) = 10, 3, 7, and -1. The sum is -8 + 10 + 3 + 7 - 1 = 11. The mean is 11 divided by 5, which equals 2.2.

For (3, 119.5): 2.5(3)+112 = 119.5. For (-415, -925.5): 2.5(-415)+112 = -1037.5+112 = -925.5. For (-14, 77): 2.5(-14)+112 = -35+112 = 77. Since all individual points satisfy the equation, 'All of These' is correct.

The series represents the number of days in consecutive months starting from January: January has 31 days, February has 28 days (non-leap year), March has 31, April has 30, May has 31, June has 30, July has 31. The next month is August, which also has 31 days. The pattern follows the calendar month length sequence.

The letters represent the first letters of the lyrics to 'The Twelve Days of Christmas': Partridge, Pear, Turtle, French, Golden, Geese, Swans, Maids, Ladies, Lords... G (Golden), B (Birds/Blackbirds), A (A-laying), L (Lords), T (Ten), I (I-something/In). However, the sequence is actually 'G' (Golden), 'B' (Birds), 'A' (A-laying), 'L' (Leaping), 'T' (Ten), 'I' (I-something). The next is 'L' for Ladies Dancing.

This band puzzle uses initials of Pink Floyd members across different eras. Position 1: P(ink) - Mason; Position 2: G(ilmore); Position 3: J corresponds to an alternate lineup member; Position 4: R(ogers/Rick); Position 5: S for Syd Barrett, the original frontman. The pattern maps letters to band member names at different positions in the sequence.

Jesus had 12 apostles. There are 7 days in a week. The sum of 12 and 7 is 19.

There are 4 seasons in a year (spring, summer, autumn, winter) and traditionally 7 Wonders of the Ancient World. Adding these gives 4 + 7 = 11. The key is knowing the standard count of wonders - not the new Seven Wonders of the World list, but the classical 7 wonders.

A tripod has 3 legs. A bicycle has 2 wheels. Adding 3 and 2 equals 5.