IKMC 2027 - Grades 5-6 - Self-study SET 10 - Thematic Batch 3

The diagram shows the roads from Alex’s home to school. How many shortest routes can he take?



Road model: use the uniform rectangular street grid shown; shortest routes never backtrack.

Goffin the parrot always lies on Sunday, Monday and Tuesday and tells the truth on the other days. Plam always lies on Thursday, Friday and Saturday and tells the truth on the other days. Today both say, “Tomorrow I will lie.” What day is today?

Two walkers start at A and B, 30 km apart, and walk towards each other. They meet after 5 hours. The walker from B covers 2 km less per hour than the walker from A. What are their speeds?

An ice-cream shop offers eight different flavours. Customers choose a combination of two flavours. How many different two-flavour combinations can the shop make?



Combination convention: use two different flavours and do not distinguish their order.

Ms Alice’s class has 32 students. She brings coloured pencils for an art lesson. First she gives one pencil to every student in group I. Then one third of the remaining students each borrow three pencils, leaving none of her pencils. Fortunately, many students brought their own, and the lesson goes well. How many pencils did Ms Alice bring?

At a pizza party, boys and girls are paired. In each pair, the boy eats either three times as many pieces as the girl or one third as many. Which listed total number of pieces could have been eaten?



Counting convention: each person eats a whole number of the given pizza pieces.

Simon weighs a peach on a two-pan balance using weights of 1 g, 3 g, 9 g, 27 g and 81 g. He places the peach on one pan and adds weights until the balance is level, finding that the peach weighs 61 g. Which weights are on the peach’s pan?

The rectangular block shown is assembled from three pieces. Each piece consists of four identical cubes and has a single colour. What is the shape of the white piece?

A shop offers strawberry, vanilla and chocolate ice creams. Sam wants two ice creams of different flavours, one for himself and one for his younger sister. How many choices are there?



Source-intended convention: a choice includes which flavour Sam receives and which his sister receives; an unassigned pair alone would give 3.

Linda has just turned 6, which is one seventh of her mother’s age. What will their combined ages be ten years from now?

Mr Brown has a rectangular field. If its length is increased by 10% and its width decreased by 10%, how does its area change?

Tony can remove a right-angled corner of a square card with one straight cut, as shown. He makes three cuts, each removing one of the square’s original corners. How many sides does the resulting polygon have?



Diagram convention: the three original corners are trimmed separately; cuts do not erase earlier cut edges or entire original sides.

Three monkeys start cycling together around a closed track. They complete exactly 5, 7 and 3 laps per minute respectively. How many minutes after the start will all three first meet again at the starting point?



Starting convention: all three begin together at the same marked starting point.

Bin draws an n×n square grid and colours the cells on both main diagonals. He colours 9 cells altogether. What is n?

Children in Kangcity enjoy card games. An exchange machine offers two operations: I, exchange one white card for four red cards; II, exchange one red card for four white cards. Alila starts with four white cards. After exactly 11 exchanges, he is said to have 31 cards in total. How many of those cards are red?



Original source choices (retained):

A. 21

B. 17

C. 14

D. 27

E. 11

Source consistency task: test the stated total after 11 exchanges. Then separately examine what happens if 31 is corrected to 37, including whether the red count is uniquely determined. Original wording and choices are retained.

A group of Kangaroo friends has a combined age of 32 years. Two years later their combined age will be 50. How many members are in the group?



The same group members are counted at both dates.

The diagram shows a rectangular grid inscribed in a circle with diameter AB=10 cm. What is the perimeter of the bold arrow shape?



Use the four equal rows and four equal columns shown; the grid cells are congruent rectangles.

A pond contains clusters of lily pads, with one frog on each pad. Pads in the same cluster may or may not be adjacent. In Figure 1, pads 2, 4, 6, 8 are adjacent to pad 5. At a thunderclap, each frog jumps to one adjacent pad in its own cluster; for example frog 5 can jump only to 2, 4, 6 or 8. Several frogs may land on one pad. First examine the four-pad cluster in Figure 2 and find the greatest number of empty pads.

1. For the six-pad cluster shown, find the least number of occupied pads and hence the greatest number of empty pads.
2. For the ten-pad cluster, find the greatest number of empty pads.
3. For the fifteen-pad cluster, find the greatest number of empty pads.
4. For the twenty-five-pad cluster, find the greatest number of empty pads.











Neighbour convention: only immediately horizontal or vertical pads are adjacent, matching Figure 1; diagonal jumps and staying in place are excluded.

Each circle in the number tower is filled with a different number from 3 to 9. The three numbers on every straight line have the same sum. Which number goes at the top?

The pipe system shown leads into tanks X and Y. Water is divided equally among the outgoing branches at every split. If 100 litres enters the system, how many litres reach Y?