# Misc Problems

### Unit Shift

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?