¿ìè¶ÌÊÓƵ

#55 Squares in squares

Set by Alison Kiddle

I have 72 mosaic tiles, which I have formed into a square picture frame. The frame measures 9 by 9, and it can display a picture of size 3 by 3.

Can you use the same tiles to create a square picture frame of a different size that holds a bigger photo?

For square frames like this more generally, with a square photo in the exact centre, how many tiles might you need? What properties do these numbers share?

Given a number of tiles, can you determine which possible frames can be made?

Solution next week

#54 New year, new numbers

Solution

1-10 can be made as:

1 = -2 + 0 – 2 + 5,

2 = 2 + (0 × 2 × 5),

3 = -2 + (0 × 2) + 5,

4 = 2 + 2 – (5 × 0),

5 = 2 + 0 – 2 + 5,

6 = (2 ÷ 2) + 5 + 0,

7 = (2 × 0) + 2 + 5,

8 = -2 + 0 + (2 × 5),

9 = 2 + 0 + 2 + 5 and

10 = (2 × 0) + (2 × 5).

Solutions for 11-30 (with the digits in order) can be found using 2⁰ and 0! in places to make 1, and joining digits into two-digit numbers to make 20 and 25.

To satisfy the order requirement, the solutions given for 4 and 6 can now be made as 2 – 0! – 2 + 5 and 2 + 0! – 2 + 5, respectively.