Puzzle of the Week #563 - Powers

a+b = s

a*b = p

What is a^4 + b^4 in terms of s and p?

Puzzle of the Week #562 - Zig Zag Sequence

I’ve constructed a sequence thusly:

I’ve started with the natural numbers 1,2,3,4,5,6,7,8,9,10 etc.

I’ve kept the first number as it is, then reversed the order of the next two, then reversed the order of the subsequent three, then reversed the order of the next four etc:

1,3,2,6,5,4,10,9,8,7,15,14,13,12,11,21,etc

Your task is to construct a formula that will instantly give you the nth term in this crazy sequence.

Puzzle of the Week #561 - Four Circles in a Triangle

Four circles are tangent to a triangle and to one another as shown. If a fifth circle is drawn enclosing the second and third circles, that circle is the same size as the fourth circle in the triangle.

If the smallest circle has radius 1, what is the radius of the second smallest circle?

Puzzle of the Week #559 - Minimal Combinations

I have 2k items and I select k of them. There are N possible combinations that I could have selected. N is a multiple of 1000. What is the smallest k could be?

Puzzle of the Week #558 - Isosceles Trapezoid

Here is an isosceles trapezoid, a shape with two parallel sides and the other two sides are equal. I have drawn two lines within the shape to split it into three regions of equal area. What is the overall area of the trapezoid?

Puzzle of the Week #557 - Palindromic Numbers

I learned the interesting fact the other day that any natural number, however large, can be written as the sum of at most three palindromic numbers. For instance, 7369 = 7227+131+11.

 Can you write the ten digit number 7,275,640,031 as the sum of three palindromic numbers?

 Single digit numbers count as palindromes (eg, 7) but numbers with leading zeroes do not (eg 0330).

Puzzle of the Week #556 - Triangle and Semicircle

In this figure, what is the height of the triangle?

Puzzle of the Week #555 - Regular Nonagon

Here is a regular polygon with nine sides. If two of the regions have areas 3 and 2 as shown, what is the area of the third region?

Puzzle of the Week #554 - Fiftieth Birthday

If a person is born at a random point in a 400 year period, roughly what is the probability that their fiftieth birthday will be on the same day of the week as they were born?

Puzzle of the Week #553 - Unknown Primes

(2+1/a)(2+1/b)(2+1/c) = d

 a, b, c and d are all prime numbers, what are they?

Puzzle of the Week #552 - Five Answers

This set of equations relates unknowns a, b and c. There are five possible sets of solutions for the values of (a,b,c). Can you find them all?

a*b + c = 12

b*c + a = 12

c*a + b = 12

Puzzle of the Week #551 - Seven Bricks

Seven identical bricks fit exactly in a circle as shown. What is the proportion of the length to the height of each of the bricks?

Puzzle of the Week #550 - Three Squares and a Circle

Three squares of areas 968, 722 and 2888 respectively coincide at one corner as shown.

A circle is drawn through their opposite corners.

What is the area of this circle?

Puzzle of the Week #549 - Rational Area

The diagram shows five unit circles, centred on the coordinates (0,0), (1,0), (2,0), (0,1) and (1,1).

The area of each unit circle is pi, which is irrational. The area of each of the 17 regions is also irrational, with the exception of one. Which region has a rational area?

Puzzle of the Week #548 - Fifty Pebbles

You have a bag that contains fifty pebbles, twenty-five painted white and twenty-five painted black.

You randomly remove two pebbles.

If either of them is white, discard it and put the other pebble (whether it is black or white) back into the bag.

If both are black, discard both of them.

You repeat this process until there is only one pebble in the bag, what is the probability that the final pebble is white?

Puzzle of the Week #547 - Fourth Triangle

A Pythagorean triangle can never have a 22.5 degree angle, since tan(22.5) is 1-sqrt(2) which is irrational, however a 5,12,13 triangle comes very close. (It isn’t surpassed in accuracy until it is marginally improved in a triangle whose shortest side is 3648).

If three triangles with sides in the ratio 5,12,13 are placed with their smallest angles together as shown, what Pythagorean triangle has the necessary angle to complete the right angle?

Puzzle of the Week #546 - Integer Lengths

If each of the line segments in this figure are integers, what is the length of the red line?

Puzzle of the Week #544 - Exact Expression

a, b and c are each real numbers between 0 and 1.

I have a number ‘a’ and I want to find an exact expression for it.

I take the reciprocal and get that 1/a = 1+b.

I subtract the 1 and take the reciprocal. I find that 1/b = 3+c.

I subtract the 3 and take the reciprocal. I discover that 1/c = 4+b.

What is the exact value of a?