# 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.

### Stuck in the Past

##### Age 7 to 14 Short Challenge Level:

Weekly Problem 13 - 2013
2002 started on a Tuesday. In which years between now and 2015 will each date fall on the same day of the week as it fell that year?

### Take a Leaf

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

Weekly Problem 19 - 2015
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?

### Royal Arm Wrestling

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

Weekly Problem 15 - 2007
The Kings of Clubs, Diamonds, Hearts and Spades, and their respective Queens, are having an arm wrestling competition.

### Back in Time

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

Weekly Problem 29 - 2015
How many times a day does a 24 hour digital clock look the same when reflected in a horizontal line?

### Weekly Lies

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

Weekly Problem 4 - 2010
Mr Ross tells truths or lies depending on the day of the week. Can you catch him out?

### 12345

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

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

### Almost Constant Digits

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

Weekly Problem 46 - 2010
Consider a 10-digit number which contains only the numbers 1, 2 or 3. How many such numbers can you write so that every pair of adjacent digits differs by 1?

### Suit Sequence

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

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

### 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?

### 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?

### 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?

### Halloween Day

##### Age 11 to 16 Short Challenge Level:

Weekly Problem 44 - 2015
One year there were exactly four Tuesdays and four Fridays in October. On what day of the week was Halloween.

### Collatz-ish

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

Weekly Problem 35 - 2014
A sequence is generated using these rules. For which values of n is the nth term equal to n?

### 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?

### Odd One Out

##### Age 16 to 18 Short Challenge Level:

In this short problem, can you deduce the likely location of the odd ones out in six sets of random numbers?