# Exploring and Noticing Structure - Short Problems

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Exploring and Noticing Structure.

### Royal Arm Wrestling

##### Age 11 to 14 Short Challenge Level:

How many arm-wrestles will there be in the royal arm wrestling championship?

### Suit Sequence

##### Age 11 to 14 Short Challenge Level:

A pattern repeats every six symbols. What are the 100th and 101st symbols?

### Out of Sync

##### Age 11 to 14 Short Challenge Level:

Albert Einstein could see two clocks which were out of sync. For what fraction of the day did they show the same time?

### Halloween Day

##### Age 11 to 14 Short Challenge Level:

One year there were exactly four Tuesdays and four Fridays in October. On what day of the week was Halloween.

### How Many Rectangles?

##### Age 11 to 14 Short Challenge Level:

By drawing 5 horizontal and four vertical lines, one can form 12 rectangles. What is the greatest number of rectangles that can be formed by drawing 15 lines?

### Knockdown

##### Age 11 to 14 Short Challenge Level:

Weekly Problem 51 - 2016
Pegs numbered 1 to 50 are placed in a row. Alternate pegs are knocked down, and this process is repeated. What is the number of the last peg to be knocked down?

### Take a Leaf

##### Age 11 to 14 Short Challenge Level:

An examination paper is made from five pieces of paper. What is the sum of the other page numbers that appear on the same sheet as page 5?

### Almost Constant Digits

##### Age 11 to 14 Short Challenge Level:

How many 10-digit numbers containing only 1s, 2s and 3s can you write?

### Not a Zero

##### Age 11 to 14 Short Challenge Level:

What is the last non-zero digit of $2^{57} \times 3^4 \times 5^{53}$?

### Too Close to Call

##### Age 11 to 14 Short Challenge Level:

Weekly Problem 24 - 2012
Can you put these very close fractions into order?

### Back in Time

##### Age 14 to 16 Short Challenge Level:

How many times a day does a 24 hour digital clock look the same when reflected in a horizontal line?

### Newspaper Sheets

##### Age 14 to 16 Short Challenge Level:

From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

### Collatz-ish

##### Age 14 to 16 Short Challenge Level:

A sequence is generated using these rules. For which values of n is the nth term equal to n?

### 12345

##### Age 14 to 16 Short Challenge Level:

Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?