National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 1
1. A Cretan said, All Cretans are liars. Assuming that no self-contradictory
statements are made, can he be telling the truth? Can he be the only
Cretan

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 5
1. Given any three positive integers, use the Pigeon-hole Principle to show
that there must be two of them whose sum is even. Is it true that there
must

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 6
1. Assume that a person shakes hands with another person at most once and
that a handshake between two persons is counted as one handshake.
(a) Draw a gr

Geometric series
Theorem:
n +1
o
a(r 1)
a+ar+a r + +a r =
r 1
2
n
Arithmetic series
(nk)
= nCk = nPk/k!
Number of routes on a rectangular grid: On a rectangular grid, the
number of routes from (i,j) to (k,l), where i k and j l, moving
easterly or northe

GEK1505/GEH1036
NATIONAL UNIVERSITY OF SINGAPORE
GEK1505/GEH1036-Living with Mathematics
SEMESTER 1 AY 2015/2016
Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. This assessment paper contains FOUR questions and comprises EIGHT printed
pages, including

GEK1505
NATIONAL UNIVERSITY OF SINGAPORE
GEK1505-Living with Mathematics
SEMESTER 1 AY 2014/2015
Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. This assessment paper contains FIVE questions and comprises TWELVE printed
pages, including this page.
2.

GEK1505, SEMESTER 2 AY 2015/2016
(1) (ANS: 7, 1.5, 4, No)
(i) 7 (at vertex G) (ii) The distance from A to each of the other vertices in alphabetical
order is 1, 2, 2, 1, 1, 1, 2, 2. Thus ans = 12/8 = 1.5 (iii) 4 (iv) No since there are 4 odd
vertices.
(2)

GEK1505
NATIONAL UNIVERSITY OF SINGAPORE
GEK1505 LIVING WITH MATHEMATICS
SOLUTION
(SEMESTER 2: AY 2013-2014)
Question 1 [20 marks]
(a) [12 marks]
(i) Answer: 32
Let Ai be the set of positive integers which are multiple of i. Since A9 A3 , we
have
|A3 A7

GEK1505
NATIONAL UNIVERSITY OF SINGAPORE
GEK1505-Living with Mathematics
SEMESTER 2 AY 2014/2015
Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. This assessment paper contains FOUR questions and comprises TWELVE printed
pages, including this page.
2.

GEK1505
NATIONAL UNIVERSITY OF SINGAPORE
GEK1505 LIVING WITH MATHEMATICS
(SEMESTER 2: AY 2013-2014)
Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. Please write your matriculation/student number only. Do not write your name.
2. This examination paper

GEK1505, 2014/15 sem 2
(1) (i) 5 (ii) 12/8 = 1.5 (iii) 8 (iv) D, F (v) 4. It contains a K4 and so it is 4. A
4-colouring is certainly possible.
(2) Let n be the number of vertices. Since a tree must have n 1 edges, by the Degree
Theorem,
5 + 4 2 + (n 5) =

GEK1505/GEH1036
NATIONAL UNIVERSITY OF SINGAPORE
GEK1505/GEH1036-Living with Mathematics
SEMESTER 2 AY 2015/2016
Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. This assessment paper contains FIVE questions and comprises SEVEN printed pages,
including

2014/15 sem 1
(a) The edges are chosen in the following order:
HF, HG, GB, BE, BA, AD, DI, BC, IJ
Thus the 5th edge is AB. The weight of this spanning tree is
1 + 2 + 2 + 1 + 2 + 1 + 1 + 3 + 4 = 17.
(b) Let x, y be the number of vertices of degree 4, 5 re

GEK1505, 2014/15 sem 2
(A) (a) (3,3,4,4,4,4,4,4), (b) 3, (c) 10/7 = 1.4286 The distances from H to A, B, . . . , G
are, respectively, 1, 2, 1, 1, 2, 2, 1. Thus the ans is 10/7, (d) 7, (e) True, (f) True, (g) 4.
(B) Now 5 minutes equal 300 seconds and 300

Graphing
Points (Vertices), Lines (Edges)
Walk: Start from a point, end another point. (Can repeat vertices)
Closed Walk: Start from point, end at same point
Path: Walk w/o repetition of vertices
Cycle: Closed Path
Length: No. of edges in the walk, includ

NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
SEMESTER 1 EXAMINATION 2012-2013
GEK1505 Living with Mathematics
November 2012 Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. This is an open book examination. Candidates may use calculators.

GEK Tutorial 9 Solution
[email protected],
email me if you need the solution dropbox link
March 31, 2016
[email protected], email me if you need the solutionGEK
dropbox
Tutorial
link 9 Solution
March 31, 2016
1 / 22
Review - Coding
Shift Transformation

GEK Tutorial 8 Solution
[email protected],
email me if you need the solution dropbox link
March 25, 2016
[email protected], email me if you need the solutionGEK
dropbox
Tutorial
link 8 Solution
March 25, 2016
1 / 25
Review - Coding
Representation: Deci

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 4
1. Internet smileys are used to convey the mood or facial expression of the
sender, e.g. :-) , and each consists of a sequence xyz of symbols, where x
co

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 3
1. A conguration of tiles is formed from n identical uniform square tiles of
side x mm and thickness y mm. The tiles are rigidly glued together one tile

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 2
1. Comment on the following piece of advice.
Never use any universal statement in your argument. It will always
weaken your argument.
2. For each of the

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 7
1. A child counts from 1 to 500 on the ngers of her left hand by starting
on the thumb, then moving to the next nger until she reaches the last
nger, aft

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 8
1. Find (i) the binary representation, (ii) the octal representation, (iii) the
hexadecimal representation of the (decimal) numbers 123 and 2006.
2. If 0

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 9
1. You suspect that the following cryptograms are obtained by rail fence
transpositions. Can you decipher them?
(a) ACIEERHMDS
(b) AHAIORCMNNV
(c) FININE

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 10
1. A group of 10,000 devotees prayed at the statue of Sir Stamford Raes
and each obtained a 4-digit number on which one bet $50 for the rst,
second and

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Semester 2, 2005-06
Tutorial 11
1. ABCD, DCEF , F EGH , HGIJ are squares.
B
C
E
G
I
K
A
D
F
H
J
L
(a) A at square lamina occupies the region ABCD and is moved to
oc

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 1
Solutions
1. If what he said is true, then he, in particular, is a liar and what he said
would be false - which would be a contradiction. Therefore, if n

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 2
Solutions
1. The piece of advice contains two universal statements. The rst statement
is equivalent to At all times in your argument, do not use any univ

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 3
Solutions
1. (a) x =
1
n [x
1
+ (x + r) + . . . + cfw_x + (n 1)r] = x + 2 (n 1)r.
(b) Let n be the number of slabs that can be placed without toppling
ov

National University of Singapore
Department of Mathematics
GEK1505 Living with Mathematics
Tutorial 4
Solutions
In this tutorial, students should pay attention to the formulation of a method
or strategy for counting or enumerating rather than be concerned