Caribou
Mathematics Competition
Brock University Caribou Mathematics Competition
English | Français
(905) 688-5550 ext. 3300

## Detective Work on Calcrostic Problems

Solving a calcrostic is like checking out a crime scene for clues. The following are examples, how to get clues from single lines of the puzzle.

For example, if the row, column or diagonal is

•  a × b = a
then a must be 0 or b must be 1. It is easy to find out which case applies. We look at other lines that include a and b. For example, if a=0 then  c + a = c and if b=1 then  c + b = d. Similarly, from  a ÷ b = a follows a=0 or b=1 and from each of  cd × b = cd,  cd ÷ b = cd follows b=1.
•  a + b = cd
then it follows that c=1 because the sum of two 1-digit numbers can not be more than 9+9=18 and if both are different then not more than 9+8=17. The same conclusion can be made from  cd  b = a.
•  a × b = c
then what is special is that the result is only a one digit number, so less than 10. Also, a, b, c are all different, so none of them can be 1 or 0. Thus, one of a, b must be 2 and the other one 3 or 4 and c is 6 or 8. The same conclusion can be drawn from  c ÷ b = a.
•  a × a = b
then b is a square number unequal a and b<10, so a=2, b=4 or a=3, b=9.
•  c + ea = eg
then the first digit (the tens) in ea and in eg are the same, so we conclude that  c + a = g.
•  c + ea = fg
then the first digit (the tens) in ea and in fg are different. This can only be due to a carry over, so we conclude that  e + 1 = f and  c + a = 10 + g.
Try to find more clues, for example:
• What can be concluded from  a × a = ba ? Which values can a, b only have?
• What follows from  eb × c = cd ? Which value can e only have?
• If you know that = 1 then what does  fg ÷ c = d tell you about f, c, d ?

Have fun!

Video solutions are available for the following calcrostics problems from Caribou contests:
Nov 2011, grade 7/8 question 23
Oct 2011, grade 7/8 question 17
May 2011, grade 7/8 question 17

## (on Wednesdays)

OCTOBER 23RD, 2013

## Previous Results

### Top 10

1st
Serena Zhang
Mackenzie Community School, ON
2nd
Gorden Tseng
Maple Ridge Secondary, BC
3rd
Kevin Chi
Mackenzie Community School, ON
4th
Ann Chen
Maple Ridge Secondary, BC
5th
Winston Leung
École Royal Vale, QC
6th
Thomas Jeong-ho Song
Maple Ridge Secondary, BC
7th
Annika Polert
École Royal Vale, QC
8th
Derek Maroney
Maple Ridge Secondary, BC
9th
Evgeni Maiseyeu
Maple Ridge Secondary, BC
10th
Zahra Altalibi
École Royal Vale, QC

Detailed Results

### Top 10

1st
Victor Lamoureux
Astral Drive Junior H.S., NS
2nd
Kevin Genereux
St. Ignatius S.S., ON
3rd
Jingfeng Shan
École Royal Vale, QC
4th
Jae Hyoung Hur
École Royal Vale, QC
5th
Alex Oxorn
École Royal Vale, QC
6th
Jeff Mason
Sacred Heart H.S., ON
7th
Aidan Bermingham
École Royal Vale, QC
8th
Hyunsu An
École Royal Vale, QC
9th
Rachel Ross
Astral Drive Junior H.S., NS
10th
William Goodman
École Royal Vale, QC

Detailed Results

### Top 10

1st
Chenyan Wang
Homelands Senior P.S., ON
2nd
Max Shao
École Marie-Anne-Gaboury, MB
3rd
Sonya Makarova
Centennial P.S., ON
4th
Sean Wang
Irwin Park Elementary, BC
5th
Ryan Torrington-smith
Castor Valley E.S., ON
6th
Jasper Liang
Marlborough Elementary, BC
7th
Jeffrey Dawson
8th
Sandra Nguyen
Homelands Senior P.S., ON
9th
Sumyung Jang
George Street M.S., NB
10th
David Lu
Homelands Senior P.S., ON

Detailed Results

### Top 10

1st
Jesse Li
Sir Wilfrid Laurier Elementary, BC
2nd
Aileen Zhang
Kwayhquitlum M.S., BC
3rd
Michelle Liu
Marlborough Elementary, BC
4th
Catherine Yang
Marlborough Elementary, BC
5th
William Wang
École élémentaire publique Kanata, ON
6th
Kenneth Jiang
Bedford South School, NS
7th
Boya Yang
Marlborough Elementary, BC
8th
Max Jiang
Bedford South School, NS
9th
Saranya Varakunan
Three Valleys P.S., ON
10th
Eric Leroux
D. Aubrey Moodie E.S., ON

Detailed Results

### Top 10

1st
Neel Nair
Three Valleys P.S., ON
2nd
Tobey Sherrin
Ryerson P.S., ON
3rd
Fuyang Deng
Winston Churchill P.S., ON
4th
Ethan Eastwood
Halifax Public Library, NS
5th
Raeed Zafar
Thamesville Area Central E.S., ON
6th
Kaitlyn Pereira
Laurelwood P.S., ON
7th
Fan Yang
Barrhaven P.S., ON
8th
Cristian B
Laurelwood P.S., ON
9th
Daniel P
Laurelwood P.S., ON
10th
Philip Seong
Marlborough Elementary, BC

Detailed Results