When Julian and Penelope met, one was half the others age plus seven years. Ten years later when they married, Penelope was thirty, but this time one was nine tenths the age of the other. How old was Julian? (No fractions or partial years, whole numbers only).
Last winter I found myself locked out of my house because I couldn’t remember the 5-digit code to open the garage door. I used the following facts to get inside.
1. The second and third digits add up to 9.
2. The first digit is equal to the second digit cubed.
3. The sum of the third and fifth digits is the smallest number with exactly five divisors.
4. The fourth digit is equal to 6 times the second-to-last digit.
5. None of the digits repeat.
If we label each digit a, b, c, d and e, we get the following equations.
1. b + c = 9
2. a = b3
3. c + e = 16 (since 16 is the smallest number with five divisors – 1, 2, 4, 8 and 16).
4. d = 6 * d (The fourth and second-to-last digits are the same number, meaning d must be zero to satisfy the equation)
5. b must be zero, one or two (b can’t be three because that makes a=27, which isn’t a single digit). Zero and one result in duplicate digits (00907 and 11808 respectively) so the only remaining value for b is 2, giving us 82709 and warmth.
I see much but change little,
I am firm, irresolute,
Powerful but gentle,
I can rip apart mountains,
Yet be moved by gentle stirrings,
I am valued and wasted,
I am life itself,
And I give life to others.
A tree. A tall one can see long distances but don’t change much. They are strong and powerful, and the roots of a tree can slowly tear apart a mountain. The gentle stirrings of the wind can blow their branches and leaves. Trees are valuable as they provide wood and paper, but they are also wasted. And lastly, trees, like all plants, provide us with life-giving oxygen.
A bag contains 64 balls of eight different colors. There are eight of each color (including red).
What is the smallest number you would have to pick, without looking, to be sure of selecting 3 red balls?