Counting and enumerationGrades 1–6
Five pigeons fly into four holes: some hole is bound to hold two of them. In these tasks a child proves that something must happen however unlucky the draw: two socks of one colour from a dark drawer, two birthdays in one month. It is the first meeting with reasoning about the worst case rather than the lucky one.
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If there are more pigeons than holes, some hole holds at least two. Spread them as evenly as you like: one is always left over.
“To be sure” means whatever comes out of the bag. So imagine everything going as badly as it can, and see how far bad luck can go.
When bad luck has run out, every colour drawn once and every hole filled once, the next one is bound to give what you want.
Each question asks what is sure. The answer is how far bad luck can go, and one more.
| To be sure of | Bad luck can give first | So take |
|---|---|---|
| two born on the same day of the week | one born on each of the 7 days | 7 + 1 = 8 people |
| a red ball, from 6 blue and 4 red | all 6 blue balls | 6 + 1 = 7 balls |
| two red balls, from 6 blue and 4 red | all 6 blue balls and one red | 6 + 1 + 1 = 8 balls |
Name the holes, such as colours, months or days of the week, and play the worst luck through to its end: take everything that does not yet give what the question wants. The next one is bound to. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 1–2
A jar holds 8 lemon sweets and 2 orange ones. Without looking, how many sweets must Sasha take to be sure of an orange one?
Answer 9 sweets.
Example 2 · Grades 3–4
A drawer holds 8 black socks, 6 white ones and 2 striped ones, all mixed up. In the dark, how many socks must Gleb take to be sure of two black ones?
Answer 10 socks.
Example 3 · Grades 5–6
Each cell of a 2 by 2 grid holds 1, 2 or 3. Add up the two numbers of each row, each column and each diagonal. Is it true that two of these six sums are always equal?
1 3
2 2Answer Yes, always.
MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.
The child counts on luck: “If the first two socks are black, two are enough.” The question says “to be sure”, and luck is what it rules out.
What to say
“And if you are as unlucky as can be?”
One word of the question gets lost: “two black socks” is read as “two socks of one colour”, or “one of each colour” as “one colour”.
What to say
“Read the question again. What exactly must you be sure of?”
The child plays the worst luck through and stops there, forgetting the one more that settles it, or adds one too many.
What to say
“After the worst luck, what does the next one give?”
Put socks of two or three colours into a bag and take them out with closed eyes: how many until a pair is certain? Try it several times. Luck changes, and the answer does not.
When the child says “it could happen”, ask “and could it not?”. The difference between “can” and “must” is the whole topic.
Look for holes in everyday counts together: seven days of the week, twelve months, four seasons. “There are 30 of us: are two of us sure to share a birthday month?”