Working systematically

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

Can you substitute numbers for the letters in these sums?

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Suppose you are a bellringer. Can you find the changes so that, starting and ending with a round, all the 24 possible permutations are rung once each and only once?

In this addition each letter stands for a different digit, with S standing for 3. What is the value of YxO?

Can you find squares within a number grid whose entries add up to an even total?

Can you make the numbers around each face of this solid add up to the same total?