Find a number which is multiplied by 2 when you shift the units digit to the other end of the number.

Langford Sequences

The numbers 1, 1, 2, 2, 3, 3, ... n, n are to be arranged in a sequence in such a way that for each r = 1, 2, ... n, the two r's are separated by exactly r places. An example, with n = 4, is:

4 1 3 1 2 4 3 2

Such a sequence is called a Langford Sequence (see The Mathematical Gazette (1958) p.228).

Find a Langford Sequence with n = 8.

Problem of the Year

Can you continue this list, expressing 7, 8, 9 etc in a similar way? The rules of the game are that you must use all the digits 1,9,9 and 7 in that order, and the symbols +, -, ×, ÷, square roots and brackets.

Any of the Above?

Exactly one of the following five statements is true. Which one?

All of the following

None of the following

Some of the following

All of the above

None of the above

Four Squares

Find four perfect squares, two consisting of three digits, one of two digits, and one of one digit, using each of the numbers 1,2,3,...,9 just once.

All Change

There are nine coins in South African currency:

1c, 2c, 5c, 10c, 20c, 50c, R1, R2, R5

Combinations of these coins cover all transactions, either by direct payment or by payment involving change.

For example, an item costing R1.98 can be paid with seven coins:

R1 + 50c + 20c + 20c + 5c + 2c + 1c

or by a transaction involving only two coins: a payment of R2 and 2c change.

Of all amounts under R10, which involves the largest number of coins, either as direct payment or one involving change?

Problems: Prime reciprocals

Any rational number can be expressed as a repeating decimal. The length of the repeating segment is called the period of the decimal. So 1/7 = 0.142857142857... has period 6, while 1/11 = 0.9090909... has period 2.

What prime numbers have reciprocals with period five or less? This problem can be tackled crudely with a computer, but a technology-free solution is also possible.

Swedish Problem

The following problem appeared in the Swedish magazine Elementa 4(95):

Visa att radien till den inskrivna cirkeln hos en pythagoreisk triangel (= en rätvinklig triangel med hetalssidor) är hetalsvärd.

Can you solve it?

From Foggott to Urr

The roads are all straight and level on the plains of Oblivia. Some distances in kilometers are given in the guide:

Whair

Foggott

Urr

Gawn

225

540

1296

Whair

0

585

1521

How far is it from Fogott to Urr?

German Alphametric

Z W E I
+ D R E I
---------
F Ü N F

Can you replace the letters by the digits 1,2,3,...,9 to make the addition correct? Each letter stands for a different digit.

Fivefold Flip

Seven coins are placed in a circle, with heads up. A move consists of flipping five coins in a row. Using only this move, can you make all the coins show tails simultaneously? What is the smallest number of moves needed?