MATH TIME

Alphabet assignment

1. Assign each letter a-z a distinct integer 1-26 such that the product of the letters in each word from "one" to "twentysix" is divisible by the number it represents. For example, e*l*e*v*e*n is divisible by 11. (I did this by hand.)

2. Minimize the sum of the sixteen letters used in the making of these words (efghilnorstuvwxy). (I could do this with a program, but I'll leave that to you. Whether by hand or by machine, my minimum (245) should be easy to beat.)

3. Show that there is no way to assign each letter a-z a distinct integer 1-26 such that the SUM of the letters in each word "one" to "twentysix" is divisible by the number it represents. (I did this with a program search, but it was still tricky to solve. Permutations ahoy!)

#2 Solution

#3 Solution and Python program

Dual sequences

Two sequences a_n and b_n are defined for integers n >= 1:

a_n + b_n = 1 for all n;

b_n = 0 if n is odd;

b_n = a_n/2 if n is even.

What is a_2^64?

Solution

Eight letter anagram

Not math, but still one of my favorites.

Fill in the blanks with two eight-letter anagrams to make a true statement.

_ _ _ _ _ _ _ _ is the _ _ _ _ _ _ _ _ of time.

abcdef

Find six distinct digits (none of them 0) such that a^b + c^d = e^f.

Solution

6742816

This is part riddle.

Observe 6742816 carefully. What property does it have, shared only by less than a hundred other numbers?

What is the longest number that shares this property?

Solution

Rays

Two distinct points A and B are chosen at random in the plane. A ray is drawn in a random direction from each point. Assume the lines that contain these rays are not parallel. What is the probability that the two rays intersect?

Solution

Circle hug

Draw a circle X of radius x, tangent to a line. Then draw three circles of radius y around X, such the first and second are tangent to both X and the line, and the third is tangent to all three circles. What is y/x?

Solution

Arithmetic-geometric means

a and b are integers. c and d are the arithmetic and geometric means of a and b. Likewise, e and f are the arithmetic and geometric means of c and d. Does there exist a set of these six numbers which are all integers?

Solution

Palindrome sum

Many numbers can be expressed as the sum of two palindromes. 389 for instance is 383 + 6 as well as 323 + 66.

What three-digit (non-palindromic) number cannot be expressed as the sum of two palindromes?

(Palindromes with leading zeroes [like 030 and 0550] are not allowed, and neither are negative numbers or non-integers.)

Solution

Remainder chains

Start with a pair of positive integers. Divide the larger by the smaller. The quotient and remainder of this division becomes a new pair. Divide again, and repeat until one number is zero, ending the chain.

Example: (17 6) - (2 5) - (2 1) - (2 0) creates a chain four pairs long.

What the smallest possible higher number of the pair which starts a remainder chain five pairs long? Six pairs?

Solution

Golden

Where φ is the golden ratio = (1+¬/5)/2,

φ² + φ⁴ + φ⁶ + ... + φ¹⁴² + φ¹⁴⁴ = φⁿ - φ.

Find n.

Solution