# £100 Prize Crossnumber, Issue 02

Our original £100 prize crossnumber is featured on pages 44 and 45 of Issue 02.

Correction: There is an error in clue 4D. The 13th digit is actually larger than the 14th.
Clarification: In 40A ‘divisible’ should read ‘properly divisible’. The answer is not 2.
Clarification: In 9A ‘proper factors’ should be taken to mean factors not equal to the number. 1 should be included.

### Rules

• Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. Use of Python, OEIS, Wikipedia, etc. is advised for some of the clues.
• One randomly selected correct answer will win £100. Three randomly selected runners up will win a Chalkdust t-shirt. The prizes have been provided by G-Research, researchers of financial markets and investment ideas. Find out more at gresearch.co.uk.
• To enter, submit the sum of the across clues via this form by 5 December 2015. Only one entry per person will be accepted. This competition has now ended. Winners will be notified by email and announced on our blog by 19 December 2015.

# Thoughts on the crossnumber

The deadline to enter the Chalkdust crossnumber #1 has now passed. The winners will be announced in next week’s blog post. In this blog post, Professor Kevin Buzzard shares some thoughts on the crossnumber.

In the rules we are told that there is a unique solution to the crossnumber. On the face of it this looks like an innocuous comment — a crossnumber for which this wasn’t true would be perhaps a little disappointing (or even unfair). However both existence and uniqueness of a solution to the crossnumber are not immediately obvious, and one has to hence decide what to do with this extra information. One could decide to verify it, by solving the puzzle and checking along the way that there is a unique solution. Alternatively, one could decide to use the information to help solve the puzzle! It is not clear to me if this is “cheating”. Let me give two examples to explain what I mean.

# Pandigital square numbers

A square number containing every digit from 0 to 9 exactly once.
Chalkdust Prize Crossnumber #1, 5 down

Whilst trying to answer 5 down in the Chalkdust crossnumber, I discovered that there are 87 square numbers in base 10 that use every digit exactly once (without a leading zero). This made me wonder how many square numbers have this property in other number bases.

I quickly wrote some code to get an idea of how many pandigital (without repeating digits or a leading zero) squares there are in other number bases.

# £100 Prize Crossnumber, Issue 01

Our original £100 prize crossnumber is featured on pages 34 and 35 of Issue 01.

### Rules

• Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0.
• One randomly selected correct answer will win £100. Three randomly selected runners up will win a Chalkdust pen. The prizes have been provided by G-Research, researchers
of financial markets and investment ideas. Find out more at gresearch.co.uk.
• To enter, email crossnumber@chalkdustmagazine.com with the sum of the across clues by 22 July 2015. Only one entry per person will be accepted. Winners will be notified by email by 1 August 2015.
• Entry is now closed.