Can you complete the grid such that every row and column contains the digits 1 to 6.

All squares that are connected contain the same digit.

These words can all have a letter added and then be rearranged to make a new 5-letter word, e.g. noun union NOUN + I = UNION.

The 7 added letters are an anagram of a 7-letter word. What is the 7-letter word, and what are the new words?

MAIL
EXIT
GOSH
CITY
NEWT
DENY
MINI

Your objective is to place some diagonal 'mirrors' into the grid.

If a ray of light is shone in to the grid from each of the letters, and allowed to bounce off the internal diagonal mirrors, each will exit the grid at the twin of the letter that it entered the grid. For example, a ray entering at either letter D will bounce off some mirrors and exit the grid at the other letter D.

Each row and each column will contain exactly two of the diagonal mirrors.

What is represented by this BrainBat?

ABCDEFGHIJKLMNOPQRSTUVWYZ
QQQQQQQQQQQQQQQQQQQQQQQQQ

Fill in the missing numbers with the digits 1 to 9. A diamond shape in the middle means that the four numbers around it add to 20.

Each uses a different way to add to 20, i.e. if there is already 1 + 3 + 8 + 9, then there will not be another using the digits 1, 3, 8 and 9 (in any order). The same digit isn't allowed to touch, even diagonally.