Outfits, ice cream cones, paths, handshakes and hidden shapes. Five levels of how-many-ways questions with pictures, plus a printable worksheet.
Each question asks how many different ways something can be done, such as how many outfits come from 2 hats and 3 pairs of shoes, or how many handshakes happen when 5 friends greet each other once. The skill is systematic counting: making a complete list without missing a case and without counting one thing twice. This is the first step of combinatorics, and it appears in children's maths contests because a child cannot guess it. A young child usually counts the first few cases and stops. The puzzles reward a plan instead, such as keeping the hat fixed and trying every pair of shoes, then moving to the next hat. A second idea runs through the levels: whether order matters. Lemon on top of grape is a different cone from grape on top of lemon, but a cup of lemon and grape is the same cup either way. The wording says which rule applies each time.
Level 1, Mix and match, asks how many outfits a child can make from 2 or 3 hats and 2 or 3 pairs of shoes, with four answers to choose from. Level 2, Count carefully, uses small numbers and answer choices: cups with 2 different scoops from 3 flavours, paths for a frog that hops only right or down on a small grid, and handshakes among 3 or 4 friends. Level 3, Order matters, asks for a typed number: cones where the top and bottom scoop differ, cones that may repeat a flavour, 2-digit numbers made from three cards, bigger grid paths, and handshakes among 5 friends. Level 4, Rules and shapes, adds a restriction, such as one hat and shoes pair that does not match, 2-digit numbers that cannot start with 0, and pictures where the child counts every square or every triangle. Level 5, Pattern hunter, mixes ways up 7 to 10 stairs by hops of 1 or 2 steps, frog paths on a 3 by 3 or 4 by 4 grid that must avoid a puddle, outfits from 4 hats where two hat and shoes pairs never go together, 2-digit numbers from five cards that use two different cards and must be bigger than 50, and squares in a 4 by 4 grid. Answers stay below 100. Easier counts such as 4 or 5 stairs, 6 friends shaking hands or a fan of five triangles sit on level 4.
No answer is typed in by hand. Each question is a small program that lists every case, with loops and recursion, and counts the list. The outfit question lists every hat and shoe pair and drops the forbidden one. The path question follows every route from the start to the house, choosing right or down at each corner. The square question visits every size and every position in the grid. The triangle question picks every pair of spots on the bottom edge together with the top corner. The tests then compare each count with the matching school formula, such as 3 times 2 for outfits, 5 times 4 divided by 2 for handshakes, or 14 squares in a 3 by 3 grid, on 500 puzzles at every level. For the answer choices, the wrong answers are plausible mistakes, such as adding instead of multiplying, or counting both orders of a handshake, and are never equal to the right answer.
Every question is short and plain, and the speaker button reads it aloud for a child who is still learning to read. The picture shows the objects, so the child can point at them and count on paper. A printed worksheet has 4, 6 or 8 new questions at the level you choose, on Letter or A4 paper, with the answers on the last page. The PDF is made in your browser and nothing is uploaded. This page does not teach the multiplication formulas, and it expects a child to make lists, draw lines or use fingers. It does not ask for probability or for numbers of three digits. Sit beside your child at levels 4 and 5 and ask, how do you know you found them all?
Ages 6 to 8. Level 1 outfit questions need only counting pairs in a picture. Levels 4 and 5, with rules and shapes, suit a strong 7 or 8 year old working with a pencil.
A program lists every case and counts them, then the tests compare the count with the school formula for 500 puzzles on every level. For example 5 friends give 10 handshakes and a 3 by 3 grid has 14 squares.
It depends, and the question says so. A cone with lemon on top and grape below is a different cone from grape on top and lemon below. A cup of lemon and grape is the same cup either way.
Five: four small squares and one big square made of all four. A 3 by 3 grid has 14 squares and a 4 by 4 grid has 30, counting every size.
Yes. The speaker button reads each question out loud using your device's voice, so a child who cannot read yet can still play.
Yes. Pick a level, 4, 6 or 8 questions and Letter or A4. Each print is new, with answers on the last page.