Analytical questions look long, but each one has a single trick that turns it into arithmetic. Learn the trick and the question takes twenty seconds instead of two minutes.
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Two formulas, and the reason both work is that one person gets counted twice.
🥇 The two formulas
Position from the other end = total − position + 1. In a row of 40, the 12th from the left is the 40 − 12 + 1 = 29th from the right.
Total in the row = position from left + position from right − 1. Someone 7th from the left and 8th from the right sits in a row of 7 + 8 − 1 = 14.
The −1 and +1 are there because the person themselves is counted from both ends.
Ordering puzzles: draw the line. "A is taller than B, C is shorter than B, D is taller than A." Write them left to right as you read: D > A > B > C. The tallest is D and the shortest is C.
Draw the circles. A conclusion is only valid if it must be true in every possible drawing.
⭕ What follows and what does not
| Statements | Valid conclusion | Invalid |
|---|---|---|
| All roses are flowers. All flowers are plants. | All roses are plants ✅ | All plants are roses ❌ |
| All cats are animals. | Some animals are cats ✅ | All animals are cats ❌ |
| Some pens are red. | Some red things are pens ✅ | Some pens are not red ❌ |
"Some A are B" says nothing about the rest of A. From "some pens are red" you may not conclude that some pens are not red — it is possible that every pen is red. Olympiad papers set this trap every year.
Test a conclusion by trying to break it. If you can draw one picture that fits all the statements but makes the conclusion false, the conclusion is invalid. If you cannot, it is valid.
Three counting puzzles that come up constantly.
🧊 A painted cube cut into 27
Paint a big cube all over and cut it into 3 × 3 × 3 = 27 small cubes.
The 8 corners have three painted faces.
The 12 edge cubes have two painted faces.
The 6 face-centres have one painted face.
The 1 cube right in the middle has none. Check: 8 + 12 + 6 + 1 = 27.
Clock angles. The hour hand moves 30° per hour and the minute hand 6° per minute. At 3 o’clock the hands are 90° apart, at 4 o’clock 120°, and at 6 o’clock 180°.
Days of the week. An ordinary year has 365 days, and 365 ÷ 7 leaves 1 over — so the same date moves on by one weekday each year, and by two after a leap year.
12 Olympiad questions on ranking, logic and cubes. 🍀
