The idea in brief. Turn the solid into a table of columns, one square of the footprint per column. Each column is capped by the smaller of its front-view height and its side-view height. A greatest fill is proved by filling every column to its cap and looking at both views again.
Lesson video, with the working
Least removed, most added
3 min 38 s
The film teaches 2 questions in turn. Each question shows first; its working is written as the film explains it.
Solid A is built from 9 unit cubes standing on a table. More unit cubes are added to it, so that every cube still rests on the table or on another cube. What is the greatest number of cubes that can be added without changing the front view or the side view?
Cubes added = 13 − 9 = 4
Ans: 4
Example 1
Solid B is built from 22 unit cubes. Some of the cubes are taken away, so that the solid that is left has the front view and the side view shown, and every cube left still rests on the table or on another cube. What is the least number of cubes that must be taken away?
Cubes taken away = 22 − 16 = 6
Ans: 6
How to solve it
Lay the footprint out flat. Each square is one column of cubes, so write each stack's height in its square.
Read the two views you must keep: the front view is 3, 2 and 2, and the side view is 3, 1 and 1.
Each column's cap is the smaller of its two view heights, such as 3, 2 and 2 along the front row.
Fill every column to its cap. The row totals are 7, 3 and 3, so 13 cubes, and both views are unchanged.
Subtract what was there: 13 minus 9 is 4, the greatest number that can be added. For least removed, leave the most.
Each square of the footprint is one column. Its cap is the smallest height the views allow. The greatest fill is the one that leaves both views unchanged.
Practice questions with worked solutions
Question 1 is free to try. The other 5 open in the free trial, each with its full working.
Question 1 of 6 · Basic · free to trySolid A is built from 9 unit cubes. Looking at it from the front, three…Solid A is built from 9 unit…
Solid A is built from 9 unit cubes. Looking at it from the front, three columns of squares are seen, 3 squares high, 2 squares high and 2 squares high. How many of the 9 cubes does the front view not show?
How do you find the most cubes added without changing the views in P6?
A solid of unit cubes stands on a table, and the question asks how many cubes can be added, or must be removed, so that its front, side or top views stay the same.
What your child learns here
A solid of unit cubes stands on a table, and the question asks how many cubes can be added, or must be removed, so that its front, side or top views stay the same. It builds on building solids with unit cubes and drawing their views from earlier years, used here as a Primary 6 reasoning question.
Read each square of the top view as one column of cubes standing on the table.
Read the front view and the side view as the tallest column in each line of the solid.
Cap each column at the smaller of its front view height and side view height.
Find the greatest number that can be added by filling every column to its cap and subtracting what is there.
Find the least number to remove by keeping as many cubes as the views allow.
Does the answer make sense?
No column may stand taller than the front view or the side view shows for its line, so a column's height is capped by the lower of the two. When cubes are added or removed, your child can redraw both views from the final column heights and see that they match the ones given.
How to practise these questions
Basic questions train that a view shows a column, not every cube inside it. Intermediate questions add matching two views to find the height of each column. Challenge questions ask for the fewest cubes removed or the most cubes added while the views stay the same. When the views and the cubes do not agree, read the written step-by-step solution.
Questions parents ask
What is the front view of a solid? It is the flat shape you see when you look straight at the solid from the front, at eye level. Each square in it stands for a column of cubes seen from that side, however many cubes are hidden behind.
How do you find the most cubes added without changing the views? Give each column a cap equal to the smaller of its front view and side view heights, fill every column to its cap, and subtract the cubes already there.
Is this topic in the P6 syllabus? It applies outcomes from earlier years: building solids with unit cubes (Primary 5) and drawing 2D representations of cubes and cuboids (Primary 4). This page teaches it as a Primary 6 reasoning question.
