Difftriangles & Sumtriangles
DiffTriangles
These puzzles have triangles and squares that share sides. A triangle always has exactly two squares on its sides – the remaining side has either a triangle or is empty. A triangle’s number is the difference of the two adjoining squares.
SumTriangles
These puzzles use addition in place of subtraction. The value of a triangle is the sum of its two or three square neighbors.
The challenge
Fill in the missing numbers to make each type of puzzle work.
How to create
Making puzzles without loops is easy. Draw an alternating sequence of squares and triangles. Then put in numbers starting at one end working your way to the far end. When you are done, remove some of the numbers. Look at the Bonus Material for ideas on how to make these puzzles with loops.
Bonus Material
DiffTriangles
DiffTriangles puzzles have triangles and squares that share sides. A triangle always has exactly two squares on its sides, and the remaining side has either a triangle or is empty. A triangle’s number is the difference of the two adjoining squares. The challenge is to supply the missing numbers.
Constructing Puzzles Making puzzles without loops is easy. Draw an alternating sequence of squares and triangles, put in numbers starting at one end, and then work your way to the far end. When you are done, remove some of the numbers. Making puzzles with loops or more complicated interactions is trickier; however, the effort pays off with some challenging puzzles!
When your child gets very comfortable with these, they may want to take a turn creating some new puzzles of their own. They should have fun and learn a lot by figuring out how the numbers fit together.
Strategies for Solving The places to do first are any triangles between two filled in squares. Another easy case is a square next to a filled triangle that has a smaller filled square next to it – in this case, because we are not working with negative numbers, there is only one choice for filling in the empty square. The most common case is a square that has two possible values looking in one direction, and two other possibilities looking in the other direction – there is usually only one number that overlaps in those possibilities.
Here are some examples with lots of interconnections.
SumTriangles
SumTriangles puzzles are just like DiffTriangles only they use addition in place of subtraction. The value of a triangle is the sum of its two or three square neighbors. Make these puzzles using methods similar to DiffTriangles. SumTriangles puzzles are typically simpler to solve than DiffTriangles.
Helping your child
Puzzles are meant to be challenging and to take time, so please don’t ruin the fun by telling your child how to do them. These puzzles are chosen so that you can create them easily and then have fun solving them together.
If your child gets stuck on a puzzle, you have several options. You can, of course, give very small hints, if you can think of things that won’t give away the puzzle. You can suggest looking at smaller or simpler versions of the puzzle. Encourage your child to be bold in their ideas, even if sometimes they lead to dead ends. We all learn a lot from our mistakes and dead ends! Let your child know that it is perfectly okay not to solve a puzzle on the first (or second or third) try, and that useful ideas may occur to them if they leave the puzzle alone for a day or two.
These puzzles are meant to be fun and to teach problem solving. One of the greatest mathematical pleasures is that AHA moment, after many false starts and much wrestling with a problem, when the answer is finally discovered – be sure to let your child experience that feeling of discovery as many times as you can!