L&NDeltaTalk

History Numbers game How many trees? Algorithm Lily vs machine L&N assignment Conclusion
Letters & Numbers: A Vehicle to Illustrate
Mathematical & Computing Fundamentals
Lighthouse Delta 2013
Steve Sugden, QUT
Phil Stocks, Bond University
28/11/2013

SBS Letters & Numbers
Letters & Numbers is a popular game show on SBS
Fairly new to Australia, but not to Europe
Based on the French des Chi¤res et des Lettres (similar to UK
Countdown)
According to Wikipedia "the oldest TV programme still
broadcast on French Television and one of the longest-running
game shows in the world."
Hosted by Richard Morecroft, David Astle (cryptic crossword
writer) & Lily Serna (UTS maths HDR student)

Anagrams
In the 1990s at Bond I taught the Data Structures subject
about 15 times (MODULA-2, Oberon-2, Delphi)
One of the topics I included was bag - a generalization of set
where elements can occur more than once
Implement by an array of cardinals (frequencies)
Index type of array maps to the set of possible bag elements
For example, a bag of characters (letters) can be used to
check for anagrams or sub-words of a given string
Newspaper anagram-type games are common: The
Australian: "Circuit Breaker", Courier-Mail: "Jumble", GC
Bulletin: "Focus", etc.
Just what is needed for the Letters part of L & N

Letters game in Excel/VBA
I have a strong preference for the Wirth family of languages,
but these are out of vogue nowadays
Not a huge fan of VB, but given my extensive work with
Excel, it made sense for me to get up to speed with VBA
Was not a very di¢ cult task to implement the bag abstraction
in VBA using a frequency array
The code simply computes the bag of the entered word, and
checks every word in the dictionary, looking for sub-bags

Example
Bag for the "word" RETAIONBE:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
1 1 0 0 2 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 0
Some words whose bags are sub-bags of this are BARITONE,
REOBTAIN (the bags for these are equal, i.e., the words are
anagrams):
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 0

Numbers game
Given six numbers (operands) and a target number, we seek
an expression which evaluates to the target, and includes at
least two of the six inputs (operands) plus any number of
instances of +, , , /
Operands can be used without replacement (once only) but
operator use is unrestricted
Parentheses may be inserted wherever required
Example: Make the target 547 from a subset of the numbers
4, 7, 8, 10, 50, 75
Solution: 547 = (10 50) + (4 10) + 7?
Is this correct?

A more interesting challenge
Solving the "numbers" part of the game on a computer is
more challenging
Arithmetic expressions are not linear like strings
They are trees with a recursive structure

Binary tree
A binary tree is usually de…ned in a recursive fashion
Tree ::= empty j (LTree, Root, RTree)
In other words, a binary tree is either empty (nothing at all) or
it consists of a sequence of three objects: the left (sub-)tree,
the root node, the right (sub-)tree
This is a typical BNF (Backus-Naur) recursive syntax
de…nition, and may be compared mathematically to a
recurrence relation

Arithmetic expression tree
"No-one, I think, is in my tree - I mean it must be high or low"
(John Lennon, Strawberry Fields Forever, 1967)
Binary tree associated with an arithmetic expression (AE)
internal nodes: operators
external nodes: operands
Example: AE tree for (2 (8 1) + (3 4))
+
××
−2
8 1
3 4

In-order tree traversal
PROCEDURE Traverse(t : Tree);
// This mode of traversal generates a normal (in-…x) AE.
// Parens necessary to avoid ambiguity: (2 (8 1) + (3 4))
BEGIN
IF t is not empty THEN
Traverse left subtree of t
Process root node of t
Traverse right subtree of t
END
END Traverse;
3
1
2
5
6
7 9
8
4
+
××
−2
8 1
3 4

Post-order tree traversal
PROCEDURE Traverse(t : Tree);
// This mode of traversal generates a post…x AE
// Parens unnecessary: 281 34 + (HP calculators)
BEGIN
IF t is not empty THEN
Traverse left subtree of t
Traverse right subtree of t
Process root node of t
END
END Traverse;
2
1
5
3
9
6 7
8
4
+
××
−2
8 1
3 4

Operands & rules
Operands may be used only as frequently as they appear
All intermediate results must be positive integers
On the SBS program, operands are split into large and small,
but this does not a¤ect our approach
Operands do not exceed 100 and target is more than 100
this means that at least two operands must be used

Reaching the target
Target may be impossible to reach
Some targets may be reached in many ways
# possible expressions is huge (see later)
Intermediate results could over‡ow or div zero might occur
code to generate trees must check for these
Many trees are equivalent (they generate the same target)
Other equivalences –commutative & associative operators

Catalan number
For n operands we have exactly n 1 operators
Tree has n operand nodes (interior nodes) plus n 1
operators (leaf nodes), giving 2n 1 nodes in total
Start with “interior tree” generated by the n 1 interior
operator nodes
The number of these is given by the Catalan number
Cn 1 =
(2n 2)!
(n 1)!n!

# operators
There are four possible operators and these may be chosen
n 1 times with replacement
Thus, the number of choices here is 4n 1

# operands
Now graft n operand nodes onto this tree
Since order matters and operands can only be used once, the
number of ways of doing this is the number of permutations
of n objects chosen from 6
This is
6!
(6 n)!

The number of trees
The multiplication rule now gives us the total number of trees
with n operands, denoted Tn
Tn =
(2n 2)!
(n 1)!n!
4n 1 6!
(6 n)!
This expression ignores the possibilities of semantically
equivalent trees
Also ignores unsuitable trees such as those which generate
division by 0 or improper …nal or intermediate results
Examples: “5 7” or “25/4” or “10/(3 3)”
Code checks for these and only builds proper trees

# possible expression trees
Simplifying the quotient Tn/Tn 1 yields a useful recurrence
Thus, the number of trees may also be expressed recursively
as T1 = 6 and
Tn =
8 (2n 3) (7 n)
n
Tn 1 if 2 n 6
n Tn
1 6
2 120
3 3, 840
4 115, 200
5 2, 580, 480
6 30, 965, 760

Algorithm
How to generate a forest this large?
We need to generate all possible expression trees with n
operand nodes and n 1 operator nodes (total 2n 1 nodes)
for 2 n 6
Tree with just one node (one for each of the original 6
operands) give us initial nursery
At any stage, to get all trees with m operand nodes, we graft
together all possible combinations of trees with operand node
counts adding up to m
Need to keep track of which operands have already been used
(cannot use an operand twice)

Create trees by grafting old ones
We need to partition the nodeset into 2 parts
For example, to get all trees with 6 operand nodes we graft:
All trees with 1 node to all with 5 nodes, and
All trees with 2 nodes to all with 4 nodes, and
All trees with 3 nodes to all with 3 nodes

Lily vs machine
L&N Episode; time 3:15
Contestant gets 1 o¤ target
Lily …nds exact answer using 5 operands
Excel model …nds exact answer 2 or 3 sec using 4 operands
ditto for Delphi model, which takes about 100 milliseconds

Delphi version of number game
This was created as the Excel version is slow
It uses the same algorithm but coded into Object Pascal
It runs about 100 times as fast
Much better for the harder problems

Letters and Numbers as a student assignment
In Bond University’s (now defunct) Bachelor of IT, there was
just one mathematics unit, known as Analytical Toolkit
It consists of a typical set of introductory discrete
mathematics topics, ending with a few weeks of very basic
introduction to probability and statistics
At Bond, Sem 1 of 2012, we set an assignment for the
students based on L & N
Mathematical background of a typical student is very poor
Also, most students had no programming experience
How could we present enough background material for the
students …rstly to understand the problem, and secondly to
implement some kind of a solution?

A reduction of problem scope
We:
reduced the scope of the problem by relaxing the Numbers
Game to 3 operands instead of 6,
cast the problem into an environment where at least some of
the solution logic may be expressed without having to write
code, and
supplied the class with some skeleton code which outlines an
overall solution strategy
These were achieved by putting a reduced version of the
problem into Microsoft Excel 2010 with VBA code and Excel
formulae and tables
For the Letters part of the problem the class was supplied
with public domain word lists, again in Excel
A session on elementary VBA was taught
Based on student feedback, this exercise was a success

Student feedback on the assignment
Although the class was small and the number of responses
even smaller, feedback was uniformly positive
"Very interesting assignment for applying algorithms to solve
problems"
"Found it to be quite fun and enjoyable however due to my
programming background it was easy."
"Although there was a steep learning curve I found this
assignment very interesting"

Conclusion
The games from Letters & Numbers are shown to be useful
vehicles for showcasing fundamentals of computation and
mathematics
A wide variety of typical CS & Discrete Mathematics concepts
in a degree program apply to the numbers game, making it a
useful teaching vehicle with wide application
Requirements for solving the Numbers Game strongly a¢ rm
why Mathematics is important in a CS or IT degree
We need more maths ambassadors/evangelists to early grades,
where lifelong attitudes to maths are usually formed
We need more people like ***Lily Serna***, an excellent role
model for young girls in particular

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