unusual square roots

simplify these square roots:

are there other examples?
what is a general rule?

what are the problems with some (many) of these?
[the denominator for 5 is 24 and this simplifies as a surd since 4 is a factor
14 is the next number after those above that doesn't have surd parts that simplify]

four triangles

the Puntmat team

has provided a more colourful version of these 11 solutions:

and a good (coloured) pdf copy of the square for printing

the Puntmat team also found two further quadrilaterals
bringing the total to 13:

domino spots - evincing a sprightlier attitude

how many spots ('pips') are there altogether on a 1 to 6 set of dominoes?

one neat aspect of this problem is the variety of approaches that could be adopted


















FC Boon elaborates his use of this problem in 'puzzle papers in arithmetic' (1937)
he argues for an inclusion of solving such problems in a maths curriculum:
"Matter and methods outside, and even beyond, the scope of the syllabus stimulate interest and widen ideas and pupils evince a sprightlier attitude which quickens their pace in more formal work."

with this arrangement, can you find the sums:

(a) 3(0 + 1 + 3 + 6 + 10 + 15 + 21) by using columns? 
     why does each column sum to 3 times a triangular number? 

(b) 6(7 + 6 + 5 + 4 + 3 + 2 + 1) from the diagonals?

(c) 8(0 + 1 + 2 + 3 + 4 + 5 + 6) by looking at the occurrences of each number?

generalising
to dominoes with spots from 1 to 'n'
can lead to a rule or rules for summing the triangular numbers

powers of 4

create a shape

fit 4 of these together

do this again
and again and again...

balloon 3D fractal

sphynx inflation

using 11

also from 'puzzle papers in arithmetic' by FC Boon

creating equality

this problem is adapted slightly from 'puzzle papers in arithmetic' by FC Boon, first published in 1937

area with perimeter

tough algebra

jumping

this could be a construction question but is probably better (more accurately) done with a computer

start with any three points

start anywhere and jump ('leapfrog') over A - the same distance the other side of it (i.e. reflect in the point  'Geogebra' enables you to do this)

then, from where you end up, jump over B

and then C

keep on jumping, over A then B then C
what happens?

James Tanton explains what happens with this problem, using (kind of) vectors

how can you return to the start after just one 'cycle' - set of three jumps?

Diffy

an intriguing task

a ppt is here

with lots of subtracting to be done
and the tasks provide a sensible reason for introducing algebra (unlike life in general...)

start with any four, smallish, numbers - in any order, initially
calculate the positive difference between an adjacent pair of numbers, looping back to the start for the right-hand end number, to produce a new set of 4 numbers
keep on doing this, line by line
until you have a good reason to stop

Herbert Wills analysed this task in 1971 and gave it the name 'Diffy'

the work was brought to my attention by a fine booklet from 'Motivated Math Project' by Stanley Bezuska at Boston College Mathematics Institute (published in 1976, I think)

you can return to the basic task of doing a 'Diffy', year on year, with fresh and increasingly complicated starting four numbers - chosen as consecutive terms from various 'standard' number patterns (see below)

various algebraic skills can be practised for ever more complicated number patterns to establish a generalisation for the number of steps it always seems to take to reach 0 0 0 0 

all starting arrangements of four numbers reduce to 0 0 0 0, quite quickly in most cases - usually in fewer than 7 steps
it's easy to make errors and tedious to check, so it can be helpful to set up a spreadsheet in advance
(using abs(difference between cells))

to begin the task(s):
ask students for any 4 numbers (not too big and not in any order) and then go through the 'diffy' process, without explanation - they try to sort out what the rules for constructing next lines are...

it is quite hard to find a set of numbers that involves more than six steps (iterations)
but it is possible

here are two examples

after a while playing around with any four numbers trying to better the "class (world) record" diffy

start to input four consecutive terms of a sequence and explore what happens
e.g. for a constant difference pattern:

following a sequence of lesson steps:

• try out several particular examples
• see what patterns are common to all the examples (or a few at least)
• decide how many steps a 'diffy' seems to take for a particular number pattern
• prove this using algebra

at various stages (maybe years) , the work can involve:

• consecutive multiples (start with a number keep multiplying by e.g. 2)
• a linear rule: start with a number, multiply by e.g. 3 and e.g. subtract 2 each time
• consecutive fibonacci numbers
• consecutive square numbers
• consecutive triangular numbers
• consecutive cubes
• consecutive terms of a general geometric sequence

these are all included on a powerpoint

Puntmat have an interesting variation of this task, using the NLVM interactive square, asking students to find a sequence of particular numbers after four iterations (steps) 

animated Penrose tiling

a very well produced video
illustrating (inflation/deflation) recursion aspects

fibonacci sequences and equations

setting up and solving linear equations
the powerpoint is here
to find the missing values in a 'fibonacci' sequence (where the next term is the sum of the previous two terms)

centred hexagonal numbers and bridge cables

a ppt is here

a quadratic generalisation

links with the difference between consecutive cubes
and this is no coincidence (see here)

this general form is also
n^3 - (n - 1)^3

summing the terms always gives a cube number

(e.g. 1 + 7 + 19 + 37 = 64 = 4^3)

 n = 4

 n = 5

 n = 6

n = 8


















centred hexagonal numbers are closely related to triangular numbers: