Puzzle of the Week #523 - Equal Sequences

These two geometric sequences have the same sum:

 

5+20+80 = 7+14+28+56 = 105

 

We can categorise each sequence by three parameters: the starting number, the common ratio and the number of terms. So the above sequences would be [5,4,3] and [7,2,4] respectively. For the purposes of this puzzle the parameters are all positive integers, the common ratio must be at least 2 and the number of terms must be at least 3.

 

Part 1: Can you find the smallest example of two sequences having the same sum?

 

Part 2: For the above sequences the number 4 appears twice in the parameters, as the common ratio of the first sequence and as the number of terms in the second sequence. Can you find the smallest example of two sequences having the same sum but where all six parameters are distinct?

Puzzle of the Week #522 - Geometric Sequence

The sum of an geometric sequence with n terms, the first of which is 1, is 127:

 1 + r + r^2 … + r^(n-1) + r^n = 127

 A subsequence that includes only every other term, but still starts at 1 and finishes at r^n totals 85:

 1 + r^2 + r^4 … + r^(n-2) + r^n = 85

 What is the sum of a subsequence that includes only every third term, but still starts at 1 and finishes at r^n:

 1 + r^3 + r^6 … + r^(n-3) + r^n = S

 

Puzzle of the Week #521 - Arithmetic Sequence

S is the sum of an arithmetic sequence with n terms:

 x_1 + x_2 + x_3 + … x_n-1 + x_n = S

 A subsequence that includes only every other term, but still starts at x_1 and finishes at x_n totals 88:

 x_1 + x_3 + x_5 + … x_n-2 + x_n = 88

 A subsequence that includes only every third term, but still starts at x_1 and finishes at x_n totals 60:

 x_1 + x_4 + x_7 + … x_n-3 + x_n = 60

 What is the sum of the whole sequence S?

 For bonus points if the first term is 0, what is the common difference between consecutive terms? 

Puzzle of the Week #519 - Word Pairs

In each couplet, the second answer is the same as the first except for the addition of one letter. If you collect the five extra letters you will spell a word.

 

A metal red or man in blue,
A blade that whirs or cuts in two.

 

A spring in stride, a triple move,
A dream you have, which time must prove.

 

Make it round, add pounds to excess,
Or squash it down with a heavy press.

 

Launch with lips, a sharp eject,
Or break apart and disconnect.

 

A scheme to mask premeditation,
Shake to wake from meditation.

 

Puzzle of the Week #512 - 4x4x4 Cube

You might have heard of the puzzle of slicing a 3cm x 3cm x 3cm cube into 27 1cm cubes, and how, even if you are allowed to rearrange the pieces between cuts, it still takes a minimum of six cuts to perform this action. There is a very clever argument that proves it.

Now consider a 4cm x 4cm x 4cm cube, cut into 64 1cm cubes. If you aren’t allowed to rearrange the pieces, will take nine cuts as shown below.

The question is: in the 4x4x4 case, with how few cuts can we slice it into 64 cm cubes if we ARE allowed to rearrange the pieces between cuts?

Puzzle of the Week #508 - Jigsaw

Place the jigsaw pieces into the grid to make a valid crossword. The eight given pieces belong in the eight outer spaces in the grid. The central square is not given: you must reconstruct it yourself. This missing central piece comprises four letters and no blanks.

Puzzle of the Week #505 - Linked Values

In this isosceles triangle, values of ‘a’ and ‘b’ are chosen such that the sides of the triangle are ab, ab and ab/2, and that the line shown going from ‘a’ away from the left  vertex to ‘b’ away from the right vertex forms a right angle. This isn’t enough information to define a and b, however if you know one of them it is possible to calculate the other.

What are all of the solutions where both a and b are integers?