January 2005
Shapes
The diagram shows three ‘U’-shaped pentominoes and five straight triominoes. The pentominoes may be arranged (without overlapping) in such a way that they can be covered by the triominoes.
Explain how.
Numbers
In a cryptarithm, the digits in a sum are represented by letters. Each letter represents a different digit, but a letter represents the same digit every time it occurs.
ABCBA = D × BE × BFFA
In this cryptarithm the five digit palindromic number ABCBA has been written as the product of its prime factors D, BE, and BFFA.
Can you work out the numbers involved? Explain the reasoning you use.
Algebra
I think of five numbers a, b, c, d, and e. When I add them in every possible pair combination (a + b, a + c, etc.) I get the numbers 0, 1, 2, 4, 7, 8, 9, 10, 11, and 12.
What are my five numbers?
(Try to use algebra rather than trial and error!)
Miscellaneous
I have four cards. On one side of each of these cards there is a single letter of the alphabet. On the other side there is single digit from 0–9. (The same letter or numeral could be on more than one card.)
I claim: ‘If a card has the letter “E” on one side, then the numeral on the other side is “4”.’
I put the four cards on the table and you see ‘B’, ‘7’, ‘E’, and ‘4’.
You want to find out whether my claim is true by turning over as few cards as possible. How many, and which, of the four cards must you turn over to find out whether the claim is true or false?