IMSO 2004 Mathematics – Taiwan Second Round – Exploration

This is a tower made from cubes (see figure). The outside part of this tower is painted. How many cubes are painted on:

(a) three sides;

(b) two sides;

(c) one side?

Mark wants to make a building, 4 cubes tall, from cubes (see figure).

(a) How many cubes does Mark need?

(b) If Mark wants to build a similar building but 8 cubes tall, how many cubes does Mark need?

The original wording uses “4 cubical tall” and “8 cubical height”; these mean heights of 4 and 8 small cubes.

Two people play a game with marbles in a box. They alternate turns. Each person can take one or two marbles at a time. The loser is the person who takes the last marble.

On your turn, how many marbles will you take to be sure of winning if there are:

(a) 6 marbles left;

(b) 8 marbles left;

(c) 20 marbles left?

(d) Change the rule: there are 100 marbles and each player can take 1, 2, 3 or 4 marbles per turn. You play first. How many marbles will you take to be sure of winning?

Explain how your strategy responds to the other player's later moves.

Two squares put on top of each other can make 9 regions (see figure).

(a) How many regions at most can be constructed from 3 squares?

(b) Using 4 squares, construct as many regions as possible with the grid transparent paper, scissors and glue. Put the result in the envelope.

(c) How many regions at most can be constructed from 4 squares?



The practical construction in (b) remains an activity. The worked example is a model to guide or check your own construction.