## Tuesday, 30 January 2018

### Problem... Square on Hexagon

#Area #CompoundArea #Pythagoras #AreaOfTriangle
#PropertiesOfShape #GeometricalReasoning #PolygonAngles

## Thursday, 18 January 2018

### Problem... Overlapping circles

#AreaCircle #Circles #Pythagoras #SectorOfCircle
#GeometricalReasoning #PropertiesOfShape

You may also find that this well-known problem, the 'Crescent Area' problem, complements work on the 'Overlapping Circles' problem well, particularly with older secondary students (follow this link for an accompanying post on the solution 'Think Through').

## Monday, 15 January 2018

### On π birthdays (2018)

All 3 year-olds, or those celebrating their 3rd birthday on or before 9 November in 2018, will turn π this year [1].  All 9 year-olds, or those celebrating their 9th birthday on or before 16 February in 2018, will turn ππ this year.  And all 36 year-olds, or those celebrating their 36th birthday on or before 15 July in 2018, will turn ππ this year [2].

If you know anyone lucky enough to be marking such a birthday this year, make sure you put it in your diary and surprise them on the day with a HapPi Birthday wish — particularly for those lucky people whose birthdays are 21 January 2015, 30 April 2008, or 26 September 1981, because they will be π, ππ, and ππ respectively on πday, 14 March [3].

T
The table below is the π-Birthday calendar for 2018.  Use the calendar to find the dob you are interested in: Look it up in the respective shaded column, then identify from the first column the date this year when they will celebrate a π-related birthday (for π birthdays first find the dob in the second column, for ππ birthdays use the third column, and for ππ birthdays use the final column).

On... you will be π you will be ππ you will be ππ
if you were born on... if you were born on... if you were born on...
01/01/2018 10/11/2014 17/02/2008 16/07/1981
02/01/2018 11/11/2014 18/02/2008 17/07/1981
03/01/2018 12/11/2014 19/02/2008 18/07/1981
04/01/2018 13/11/2014 20/02/2008 19/07/1981
05/01/2018 14/11/2014 21/02/2008 20/07/1981
06/01/2018 15/11/2014 22/02/2008 21/07/1981
07/01/2018 16/11/2014 23/02/2008 22/07/1981
08/01/2018 17/11/2014 24/02/2008 23/07/1981
09/01/2018 18/11/2014 25/02/2008 24/07/1981
10/01/2018 19/11/2014 26/02/2008 25/07/1981
11/01/2018 20/11/2014 27/02/2008 26/07/1981
12/01/2018 21/11/2014 28/02/2008 27/07/1981
13/01/2018 22/11/2014 29/02/2008 28/07/1981
14/01/2018 23/11/2014 01&02/03/2008 29/07/1981
15/01/2018 24/11/2014 03/03/2008 30/07/1981
16/01/2018 25/11/2014 04/03/2008 31/07/1981
17/01/2018 26/11/2014 05/03/2008 01/08/1981
18/01/2018 27/11/2014 06/03/2008 02/08/1981
19/01/2018 28/11/2014 07/03/2008 03/08/1981
20/01/2018 29/11/2014 08/03/2008 04/08/1981
21/01/2018 30/11/2014 09/03/2008 05/08/1981
22/01/2018 01/12/2014 10/03/2008 06/08/1981
23/01/2018 02/12/2014 11/03/2008 07/08/1981
24/01/2018 03/12/2014 12/03/2008 08/08/1981
25/01/2018 04/12/2014 13/03/2008 09/08/1981
26/01/2018 05/12/2014 14/03/2008 10/08/1981
27/01/2018 06/12/2014 15/03/2008 11/08/1981
28/01/2018 07/12/2014 16/03/2008 12/08/1981
29/01/2018 08/12/2014 17/03/2008 13/08/1981
30/01/2018 09/12/2014 18/03/2008 14/08/1981
31/01/2018 10/12/2014 19/03/2008 15/08/1981
01/02/2018 11/12/2014 20/03/2008 16/08/1981
02/02/2018 12/12/2014 21/03/2008 17/08/1981
03/02/2018 13/12/2014 22/03/2008 18/08/1981
04/02/2018 14/12/2014 23/03/2008 19/08/1981
05/02/2018 15/12/2014 24/03/2008 20/08/1981
06/02/2018 16/12/2014 25/03/2008 21/08/1981
07/02/2018 17/12/2014 26/03/2008 22/08/1981
08/02/2018 18/12/2014 27/03/2008 23/08/1981
09/02/2018 19/12/2014 28/03/2008 24/08/1981
10/02/2018 20/12/2014 29/03/2008 25/08/1981
11/02/2018 21/12/2014 30/03/2008 26/08/1981
12/02/2018 22/12/2014 31/03/2008 27/08/1981
13/02/2018 23/12/2014 01/04/2008 28/08/1981
14/02/2018 24/12/2014 02/04/2008 29/08/1981
15/02/2018 25/12/2014 03/04/2008 30/08/1981
16/02/2018 26/12/2014 04/04/2008 31/08/1981
17/02/2018 27/12/2014 05/04/2008 01/09/1981
18/02/2018 28/12/2014 06/04/2008 02/09/1981
19/02/2018 29/12/2014 07/04/2008 03/09/1981
20/02/2018 30/12/2014 08/04/2008 04/09/1981
21/02/2018 31/12/2014 09/04/2008 05/09/1981
22/02/2018 01/01/2015 10/04/2008 06/09/1981
23/02/2018 02/01/2015 11/04/2008 07/09/1981
24/02/2018 03/01/2015 12/04/2008 08/09/1981
25/02/2018 04/01/2015 13/04/2008 09/09/1981
26/02/2018 05/01/2015 14/04/2008 10/09/1981
27/02/2018 06/01/2015 15/04/2008 11/09/1981
28/02/2018 07/01/2015 16/04/2008 12/09/1981
01/03/2018 08/01/2015 17/04/2008 13/09/1981
02/03/2018 09/01/2015 18/04/2008 14/09/1981
03/03/2018 10/01/2015 19/04/2008 15/09/1981
04/03/2018 11/01/2015 20/04/2008 16/09/1981
05/03/2018 12/01/2015 21/04/2008 17/09/1981
06/03/2018 13/01/2015 22/04/2008 18/09/1981
07/03/2018 14/01/2015 23/04/2008 19/09/1981
08/03/2018 15/01/2015 24/04/2008 20/09/1981
09/03/2018 16/01/2015 25/04/2008 21/09/1981
10/03/2018 17/01/2015 26/04/2008 22/09/1981
11/03/2018 18/01/2015 27/04/2008 23/09/1981
12/03/2018 19/01/2015 28/04/2008 24/09/1981
13/03/2018 20/01/2015 29/04/2008 25/09/1981
14/03/2018 21/01/2015 30/04/2008 26/09/1981
15/03/2018 22/01/2015 01/05/2008 27/09/1981
16/03/2018 23/01/2015 02/05/2008 28/09/1981
17/03/2018 24/01/2015 03/05/2008 29/09/1981
18/03/2018 25/01/2015 04/05/2008 30/09/1981
19/03/2018 26/01/2015 05/05/2008 01/10/1981
20/03/2018 27/01/2015 06/05/2008 02/10/1981
21/03/2018 28/01/2015 07/05/2008 03/10/1981
22/03/2018 29/01/2015 08/05/2008 04/10/1981
23/03/2018 30/01/2015 09/05/2008 05/10/1981
24/03/2018 31/01/2015 10/05/2008 06/10/1981
25/03/2018 01/02/2015 11/05/2008 07/10/1981
26/03/2018 02/02/2015 12/05/2008 08/10/1981
27/03/2018 03/02/2015 13/05/2008 09/10/1981
28/03/2018 04/02/2015 14/05/2008 10/10/1981
29/03/2018 05/02/2015 15/05/2008 11/10/1981
30/03/2018 06/02/2015 16/05/2008 12/10/1981
31/03/2018 07/02/2015 17/05/2008 13/10/1981
01/04/2018 08/02/2015 18/05/2008 14/10/1981
02/04/2018 09/02/2015 19/05/2008 15/10/1981
03/04/2018 10/02/2015 20/05/2008 16/10/1981
04/04/2018 11/02/2015 21/05/2008 17/10/1981
05/04/2018 12/02/2015 22/05/2008 18/10/1981
06/04/2018 13/02/2015 23/05/2008 19/10/1981
07/04/2018 14/02/2015 24/05/2008 20/10/1981
08/04/2018 15/02/2015 25/05/2008 21/10/1981
09/04/2018 16/02/2015 26/05/2008 22/10/1981
10/04/2018 17/02/2015 27/05/2008 23/10/1981
11/04/2018 18/02/2015 28/05/2008 24/10/1981
12/04/2018 19/02/2015 29/05/2008 25/10/1981
13/04/2018 20/02/2015 30/05/2008 26/10/1981
14/04/2018 21/02/2015 31/05/2008 27/10/1981
15/04/2018 22/02/2015 01/06/2008 28/10/1981
16/04/2018 23/02/2015 02/06/2008 29/10/1981
17/04/2018 24/02/2015 03/06/2008 30/10/1981
18/04/2018 25/02/2015 04/06/2008 31/10/1981
19/04/2018 26/02/2015 05/06/2008 01/11/1981
20/04/2018 27/02/2015 06/06/2008 02/11/1981
21/04/2018 28/02/2015 07/06/2008 03/11/1981
22/04/2018 01/03/2015 08/06/2008 04/11/1981
23/04/2018 02/03/2015 09/06/2008 05/11/1981
24/04/2018 03/03/2015 10/06/2008 06/11/1981
25/04/2018 04/03/2015 11/06/2008 07/11/1981
26/04/2018 05/03/2015 12/06/2008 08/11/1981
27/04/2018 06/03/2015 13/06/2008 09/11/1981
28/04/2018 07/03/2015 14/06/2008 10/11/1981
29/04/2018 08/03/2015 15/06/2008 11/11/1981
30/04/2018 09/03/2015 16/06/2008 12/11/1981
01/05/2018 10/03/2015 17/06/2008 13/11/1981
02/05/2018 11/03/2015 18/06/2008 14/11/1981
03/05/2018 12/03/2015 19/06/2008 15/11/1981
04/05/2018 13/03/2015 20/06/2008 16/11/1981
05/05/2018 14/03/2015 21/06/2008 17/11/1981
06/05/2018 15/03/2015 22/06/2008 18/11/1981
07/05/2018 16/03/2015 23/06/2008 19/11/1981
08/05/2018 17/03/2015 24/06/2008 20/11/1981
09/05/2018 18/03/2015 25/06/2008 21/11/1981
10/05/2018 19/03/2015 26/06/2008 22/11/1981
11/05/2018 20/03/2015 27/06/2008 23/11/1981
12/05/2018 21/03/2015 28/06/2008 24/11/1981
13/05/2018 22/03/2015 29/06/2008 25/11/1981
14/05/2018 23/03/2015 30/06/2008 26/11/1981
15/05/2018 24/03/2015 01/07/2008 27/11/1981
16/05/2018 25/03/2015 02/07/2008 28/11/1981
17/05/2018 26/03/2015 03/07/2008 29/11/1981
18/05/2018 27/03/2015 04/07/2008 30/11/1981
19/05/2018 28/03/2015 05/07/2008 01/12/1981
20/05/2018 29/03/2015 06/07/2008 02/12/1981
21/05/2018 30/03/2015 07/07/2008 03/12/1981
22/05/2018 31/03/2015 08/07/2008 04/12/1981
23/05/2018 01/04/2015 09/07/2008 05/12/1981
24/05/2018 02/04/2015 10/07/2008 06/12/1981
25/05/2018 03/04/2015 11/07/2008 07/12/1981
26/05/2018 04/04/2015 12/07/2008 08/12/1981
27/05/2018 05/04/2015 13/07/2008 09/12/1981
28/05/2018 06/04/2015 14/07/2008 10/12/1981
29/05/2018 07/04/2015 15/07/2008 11/12/1981
30/05/2018 08/04/2015 16/07/2008 12/12/1981
31/05/2018 09/04/2015 17/07/2008 13/12/1981
01/06/2018 10/04/2015 18/07/2008 14/12/1981
02/06/2018 11/04/2015 19/07/2008 15/12/1981
03/06/2018 12/04/2015 20/07/2008 16/12/1981
04/06/2018 13/04/2015 21/07/2008 17/12/1981
05/06/2018 14/04/2015 22/07/2008 18/12/1981
06/06/2018 15/04/2015 23/07/2008 19/12/1981
07/06/2018 16/04/2015 24/07/2008 20/12/1981
08/06/2018 17/04/2015 25/07/2008 21/12/1981
09/06/2018 18/04/2015 26/07/2008 22/12/1981
10/06/2018 19/04/2015 27/07/2008 23/12/1981
11/06/2018 20/04/2015 28/07/2008 24/12/1981
12/06/2018 21/04/2015 29/07/2008 25/12/1981
13/06/2018 22/04/2015 30/07/2008 26/12/1981
14/06/2018 23/04/2015 31/07/2008 27/12/1981
15/06/2018 24/04/2015 01/08/2008 28/12/1981
16/06/2018 25/04/2015 02/08/2008 29/12/1981
17/06/2018 26/04/2015 03/08/2008 30/12/1981
18/06/2018 27/04/2015 04/08/2008 31/12/1981
19/06/2018 28/04/2015 05/08/2008 01/01/1982
20/06/2018 29/04/2015 06/08/2008 02/01/1982
21/06/2018 30/04/2015 07/08/2008 03/01/1982
22/06/2018 01/05/2015 08/08/2008 04/01/1982
23/06/2018 02/05/2015 09/08/2008 05/01/1982
24/06/2018 03/05/2015 10/08/2008 06/01/1982
25/06/2018 04/05/2015 11/08/2008 07/01/1982
26/06/2018 05/05/2015 12/08/2008 08/01/1982
27/06/2018 06/05/2015 13/08/2008 09/01/1982
28/06/2018 07/05/2015 14/08/2008 10/01/1982
29/06/2018 08/05/2015 15/08/2008 11/01/1982
30/06/2018 09/05/2015 16/08/2008 12/01/1982
01/07/2018 10/05/2015 17/08/2008 13/01/1982
02/07/2018 11/05/2015 18/08/2008 14/01/1982
03/07/2018 12/05/2015 19/08/2008 15/01/1982
04/07/2018 13/05/2015 20/08/2008 16/01/1982
05/07/2018 14/05/2015 21/08/2008 17/01/1982
06/07/2018 15/05/2015 22/08/2008 18/01/1982
07/07/2018 16/05/2015 23/08/2008 19/01/1982
08/07/2018 17/05/2015 24/08/2008 20/01/1982
09/07/2018 18/05/2015 25/08/2008 21/01/1982
10/07/2018 19/05/2015 26/08/2008 22/01/1982
11/07/2018 20/05/2015 27/08/2008 23/01/1982
12/07/2018 21/05/2015 28/08/2008 24/01/1982
13/07/2018 22/05/2015 29/08/2008 25/01/1982
14/07/2018 23/05/2015 30/08/2008 26/01/1982
15/07/2018 24/05/2015 31/08/2008 27/01/1982
16/07/2018 25/05/2015 01/09/2008 28/01/1982
17/07/2018 26/05/2015 02/09/2008 29/01/1982
18/07/2018 27/05/2015 03/09/2008 30/01/1982
19/07/2018 28/05/2015 04/09/2008 31/01/1982
20/07/2018 29/05/2015 05/09/2008 01/02/1982
21/07/2018 30/05/2015 06/09/2008 02/02/1982
22/07/2018 31/05/2015 07/09/2008 03/02/1982
23/07/2018 01/06/2015 08/09/2008 04/02/1982
24/07/2018 02/06/2015 09/09/2008 05/02/1982
25/07/2018 03/06/2015 10/09/2008 06/02/1982
26/07/2018 04/06/2015 11/09/2008 07/02/1982
27/07/2018 05/06/2015 12/09/2008 08/02/1982
28/07/2018 06/06/2015 13/09/2008 09/02/1982
29/07/2018 07/06/2015 14/09/2008 10/02/1982
30/07/2018 08/06/2015 15/09/2008 11/02/1982
31/07/2018 09/06/2015 16/09/2008 12/02/1982
01/08/2018 10/06/2015 17/09/2008 13/02/1982
02/08/2018 11/06/2015 18/09/2008 14/02/1982
03/08/2018 12/06/2015 19/09/2008 15/02/1982
04/08/2018 13/06/2015 20/09/2008 16/02/1982
05/08/2018 14/06/2015 21/09/2008 17/02/1982
06/08/2018 15/06/2015 22/09/2008 18/02/1982
07/08/2018 16/06/2015 23/09/2008 19/02/1982
08/08/2018 17/06/2015 24/09/2008 20/02/1982
09/08/2018 18/06/2015 25/09/2008 21/02/1982
10/08/2018 19/06/2015 26/09/2008 22/02/1982
11/08/2018 20/06/2015 27/09/2008 23/02/1982
12/08/2018 21/06/2015 28/09/2008 24/02/1982
13/08/2018 22/06/2015 29/09/2008 25/02/1982
14/08/2018 23/06/2015 30/09/2008 26/02/1982
15/08/2018 24/06/2015 01/10/2008 27/02/1982
16/08/2018 25/06/2015 02/10/2008 28/02/1982
17/08/2018 26/06/2015 03/10/2008 01/03/1982
18/08/2018 27/06/2015 04/10/2008 02/03/1982
19/08/2018 28/06/2015 05/10/2008 03/03/1982
20/08/2018 29/06/2015 06/10/2008 04/03/1982
21/08/2018 30/06/2015 07/10/2008 05/03/1982
22/08/2018 01/07/2015 08/10/2008 06/03/1982
23/08/2018 02/07/2015 09/10/2008 07/03/1982
24/08/2018 03/07/2015 10/10/2008 08/03/1982
25/08/2018 04/07/2015 11/10/2008 09/03/1982
26/08/2018 05/07/2015 12/10/2008 10/03/1982
27/08/2018 06/07/2015 13/10/2008 11/03/1982
28/08/2018 07/07/2015 14/10/2008 12/03/1982
29/08/2018 08/07/2015 15/10/2008 13/03/1982
30/08/2018 09/07/2015 16/10/2008 14/03/1982
31/08/2018 10/07/2015 17/10/2008 15/03/1982
01/09/2018 11/07/2015 18/10/2008 16/03/1982
02/09/2018 12/07/2015 19/10/2008 17/03/1982
03/09/2018 13/07/2015 20/10/2008 18/03/1982
04/09/2018 14/07/2015 21/10/2008 19/03/1982
05/09/2018 15/07/2015 22/10/2008 20/03/1982
06/09/2018 16/07/2015 23/10/2008 21/03/1982
07/09/2018 17/07/2015 24/10/2008 22/03/1982
08/09/2018 18/07/2015 25/10/2008 23/03/1982
09/09/2018 19/07/2015 26/10/2008 24/03/1982
10/09/2018 20/07/2015 27/10/2008 25/03/1982
11/09/2018 21/07/2015 28/10/2008 26/03/1982
12/09/2018 22/07/2015 29/10/2008 27/03/1982
13/09/2018 23/07/2015 30/10/2008 28/03/1982
14/09/2018 24/07/2015 31/10/2008 29/03/1982
15/09/2018 25/07/2015 01/11/2008 30/03/1982
16/09/2018 26/07/2015 02/11/2008 31/03/1982
17/09/2018 27/07/2015 03/11/2008 01/04/1982
18/09/2018 28/07/2015 04/11/2008 02/04/1982
19/09/2018 29/07/2015 05/11/2008 03/04/1982
20/09/2018 30/07/2015 06/11/2008 04/04/1982
21/09/2018 31/07/2015 07/11/2008 05/04/1982
22/09/2018 01/08/2015 08/11/2008 06/04/1982
23/09/2018 02/08/2015 09/11/2008 07/04/1982
24/09/2018 03/08/2015 10/11/2008 08/04/1982
25/09/2018 04/08/2015 11/11/2008 09/04/1982
26/09/2018 05/08/2015 12/11/2008 10/04/1982
27/09/2018 06/08/2015 13/11/2008 11/04/1982
28/09/2018 07/08/2015 14/11/2008 12/04/1982
29/09/2018 08/08/2015 15/11/2008 13/04/1982
30/09/2018 09/08/2015 16/11/2008 14/04/1982
01/10/2018 10/08/2015 17/11/2008 15/04/1982
02/10/2018 11/08/2015 18/11/2008 16/04/1982
03/10/2018 12/08/2015 19/11/2008 17/04/1982
04/10/2018 13/08/2015 20/11/2008 18/04/1982
05/10/2018 14/08/2015 21/11/2008 19/04/1982
06/10/2018 15/08/2015 22/11/2008 20/04/1982
07/10/2018 16/08/2015 23/11/2008 21/04/1982
08/10/2018 17/08/2015 24/11/2008 22/04/1982
09/10/2018 18/08/2015 25/11/2008 23/04/1982
10/10/2018 19/08/2015 26/11/2008 24/04/1982
11/10/2018 20/08/2015 27/11/2008 25/04/1982
12/10/2018 21/08/2015 28/11/2008 26/04/1982
13/10/2018 22/08/2015 29/11/2008 27/04/1982
14/10/2018 23/08/2015 30/11/2008 28/04/1982
15/10/2018 24/08/2015 01/12/2008 29/04/1982
16/10/2018 25/08/2015 02/12/2008 30/04/1982
17/10/2018 26/08/2015 03/12/2008 01/05/1982
18/10/2018 27/08/2015 04/12/2008 02/05/1982
19/10/2018 28/08/2015 05/12/2008 03/05/1982
20/10/2018 29/08/2015 06/12/2008 04/05/1982
21/10/2018 30/08/2015 07/12/2008 05/05/1982
22/10/2018 31/08/2015 08/12/2008 06/05/1982
23/10/2018 01/09/2015 09/12/2008 07/05/1982
24/10/2018 02/09/2015 10/12/2008 08/05/1982
25/10/2018 03/09/2015 11/12/2008 09/05/1982
26/10/2018 04/09/2015 12/12/2008 10/05/1982
27/10/2018 05/09/2015 13/12/2008 11/05/1982
28/10/2018 06/09/2015 14/12/2008 12/05/1982
29/10/2018 07/09/2015 15/12/2008 13/05/1982
30/10/2018 08/09/2015 16/12/2008 14/05/1982
31/10/2018 09/09/2015 17/12/2008 15/05/1982
01/11/2018 10/09/2015 18/12/2008 16/05/1982
02/11/2018 11/09/2015 19/12/2008 17/05/1982
03/11/2018 12/09/2015 20/12/2008 18/05/1982
04/11/2018 13/09/2015 21/12/2008 19/05/1982
05/11/2018 14/09/2015 22/12/2008 20/05/1982
06/11/2018 15/09/2015 23/12/2008 21/05/1982
07/11/2018 16/09/2015 24/12/2008 22/05/1982
08/11/2018 17/09/2015 25/12/2008 23/05/1982
09/11/2018 18/09/2015 26/12/2008 24/05/1982
10/11/2018 19/09/2015 27/12/2008 25/05/1982
11/11/2018 20/09/2015 28/12/2008 26/05/1982
12/11/2018 21/09/2015 29/12/2008 27/05/1982
13/11/2018 22/09/2015 30/12/2008 28/05/1982
14/11/2018 23/09/2015 31/12/2008 29/05/1982
15/11/2018 24/09/2015 01/01/2009 30/05/1982
16/11/2018 25/09/2015 02/01/2009 31/05/1982
17/11/2018 26/09/2015 03/01/2009 01/06/1982
18/11/2018 27/09/2015 04/01/2009 02/06/1982
19/11/2018 28/09/2015 05/01/2009 03/06/1982
20/11/2018 29/09/2015 06/01/2009 04/06/1982
21/11/2018 30/09/2015 07/01/2009 05/06/1982
22/11/2018 01/10/2015 08/01/2009 06/06/1982
23/11/2018 02/10/2015 09/01/2009 07/06/1982
24/11/2018 03/10/2015 10/01/2009 08/06/1982
25/11/2018 04/10/2015 11/01/2009 09/06/1982
26/11/2018 05/10/2015 12/01/2009 10/06/1982
27/11/2018 06/10/2015 13/01/2009 11/06/1982
28/11/2018 07/10/2015 14/01/2009 12/06/1982
29/11/2018 08/10/2015 15/01/2009 13/06/1982
30/11/2018 09/10/2015 16/01/2009 14/06/1982
01/12/2018 10/10/2015 17/01/2009 15/06/1982
02/12/2018 11/10/2015 18/01/2009 16/06/1982
03/12/2018 12/10/2015 19/01/2009 17/06/1982
04/12/2018 13/10/2015 20/01/2009 18/06/1982
05/12/2018 14/10/2015 21/01/2009 19/06/1982
06/12/2018 15/10/2015 22/01/2009 20/06/1982
07/12/2018 16/10/2015 23/01/2009 21/06/1982
08/12/2018 17/10/2015 24/01/2009 22/06/1982
09/12/2018 18/10/2015 25/01/2009 23/06/1982
10/12/2018 19/10/2015 26/01/2009 24/06/1982
11/12/2018 20/10/2015 27/01/2009 25/06/1982
12/12/2018 21/10/2015 28/01/2009 26/06/1982
13/12/2018 22/10/2015 29/01/2009 27/06/1982
14/12/2018 23/10/2015 30/01/2009 28/06/1982
15/12/2018 24/10/2015 31/01/2009 29/06/1982
16/12/2018 25/10/2015 01/02/2009 30/06/1982
17/12/2018 26/10/2015 02/02/2009 01/07/1982
18/12/2018 27/10/2015 03/02/2009 02/07/1982
19/12/2018 28/10/2015 04/02/2009 03/07/1982
20/12/2018 29/10/2015 05/02/2009 04/07/1982
21/12/2018 30/10/2015 06/02/2009 05/07/1982
22/12/2018 31/10/2015 07/02/2009 06/07/1982
23/12/2018 01/11/2015 08/02/2009 07/07/1982
24/12/2018 02/11/2015 09/02/2009 08/07/1982
25/12/2018 03/11/2015 10/02/2009 09/07/1982
26/12/2018 04/11/2015 11/02/2009 10/07/1982
27/12/2018 05/11/2015 12/02/2009 11/07/1982
28/12/2018 06/11/2015 13/02/2009 12/07/1982
29/12/2018 07/11/2015 14/02/2009 13/07/1982
30/12/2018 08/11/2015 15/02/2009 14/07/1982
31/12/2018 09/11/2015 16/02/2009 15/07/1982

Notes & (Select) Links:

[1] Note the use of the word 'turn' is carefully chosen here.  The problem is a useful add-on to work with students shortly after they have been introduced to π, because it leads to some invigorating discussions about the irrational nature of π, and, indeed, the nature of irrationality.  Sharing some excerpts from online discussion threads about the question is it possible to be exactly π?, such as this one, or this reddit thread, can function as potent provocations for formative argument.

[2] The generation of π-related birthdays can serve as a conceit to use with students in the development of their problem-solving skills, supporting their growing proficiency with increasingly complicated calculation, encouraging thus the development on their computational thinking.  (This problem, 'Pi on Pi day in 1592,' may also be of interest in this regard.)  Students may be asked to calculate their own π-relative birthdays, or those of their families and friends, their e-relative birthdays, their root-2-relative birthdays, etc.

[3] Also Einstein's birthday, March 14, 1879, which, incidentally, first occurs at position 74,434,701 in π.  Find where your birthday (as a string of numbers) occurs in pi here, powered by Wolfram.

## Monday, 8 January 2018

### On Frustrating Primes

7775353777 is a prime number.  It has ten digits and it looks like, on first inspection, that the second five digits reflect the first.  But a number cannot be prime and have such a property — and on closer inspection we can see that there is, indeed, no such reflection.  Such primes can be considered 'frustrating' because of the absence of this visually eye-catching, aesthetically pleasing pattern they appear, on the initial face of it, to promise.

A prime, therefore, is said to be frustrating if the first half of its digits occur again in the second half, but without a) the order repeating (i.e. abcabc, because this is divisible by 1001 [1] and thus not prime), or b) the order reflecting (i.e. abccba, because every palindrome with an even number of digits is divisible by 11 and thus not prime).

As such, a frustrating prime has an even number of digits, and contains at least six digits of which at least two are distinct [2].  Of the 68,906 six-digit primes [3], 265 are frustrating (collated in the image above).  The possible arrangements of six-digits with the first three reoccurring:

aaa···
aaa‖aaa : Not prime   divisible by 11

aab···
aabaab : Not prime  divisible by 1001
aababa : Possibly a frustrating prime (n = 10)
aabbaa : Not prime  divisible by 11

aba···
abaaab : Possibly a frustrating prime (n = 7)
abaaba : Not prime  divisible by 1001
ababaa : Possibly a frustrating prime (n = 10)

abb···
abbabb : Not prime  divisible by 1001
abbbab : Possibly a frustrating prime (n = 9)
abbbba : Not prime  divisible by 11

abc···
abcabc : Not prime  divisible by 1001
abcacb : Possibly a frustrating prime (n = 66)
abcbac : Possibly a frustrating prime (n = 58)
abcbca : Possibly a frustrating prime (n = 68)
abccab : Possibly a frustrating prime (n = 37)
abccba : Not prime  divisible by 11

The Frustrating Primes are recorded as sequence A297994 in the On-line Encyclopedia of Integer Sequences (OEIS), from which you can find this table of Frustrating Primes an for n = 1 to 10,000

The learning capital to be accrued from exploring the idea of frustrating primes, is especially evident in terms of the development of students' mathematical maturity.  The above sketch of the underlying premise of frustrating primes informs a 'thinking through' of how as teachers we could use the idea with (alongside) students, how we could approach the problem with students, structurally and pedagogically, and what questions can/should be asked of our students, should they not yet be able to ask them for themselves.

Considering the idea through this problem will encourage depth and maturation, and give us as teachers the opportunity, as I wrote about in a previous post, to 'model for [students] what it is to be mathematically mature, to behave in a mathematically mature manner when we are befuddled by the beautiful sixes and sevens of uncertainty.'   Students will not only deepen their sense of number and their appreciation of prime numbers, at the same time as being able to apply divisibility tests, consider prime factors, etc., but they may also get a kick out of doing something a little more mathematically novel than they are typically used to.

Notes & (Select) Links:

[1]  $$\left( {1000 \times \overline {abc} } \right) + \left( {1 \times \overline {abc} } \right)$$

[2]  For a two-digit number to be a frustrating prime the first half of the digits would have to occur in the second, and this would thus be a number of the form aa, which is clearly a composite number divisible by 11.  Similarly, for a four-digit number to be a frustrating prime the first half of the digits would have to occur in the second, meaning that the number would thus be of the form abab or abba, which are both composite as the former is divisible by 101 (100×ab+1×ab) and the latter is an even palindrome and thus divisible by 11.