3.4 Alphabet Test, Word Formation, Calendar & Clock
Key Takeaways
- The angle between the hands of a clock is |30H − 5.5M| degrees, so at 3:40 the hands are 130° apart.
- An ordinary year has 1 odd day and a leap year has 2, so 1 January 2027 falls one weekday after 1 January 2026.
- A century year is a leap year only if it is divisible by 400; 2000 was a leap year but 1900 was not.
- In 24 hours, the hands of a clock coincide 22 times and are at right angles 44 times.
- The word DEPOSIT has 5 letter pairs with as many letters between them in the word as in the English alphabet.
3.4 Alphabet Test, Word Formation, Calendar & Clock
Exam Focus: Short "miscellaneous" questions—letter positions, meaningful-word formation, digit conditions, calendars and clock angles—can appear as standalone questions in reasoning sections. Each takes under a minute once you know the rule, which makes them useful accuracy boosters between longer puzzle sets.
1. Alphabet Test
Position Rules
Use the EJOTY anchors (E = 5, J = 10, O = 15, T = 20, Y = 25) and the 27-sum rule for opposite letters from Section 2.2.
| Question pattern | Method | Example |
|---|---|---|
| nth letter from the left | Position n | 14th from the left = N |
| nth letter from the right | Position 27 − n | 5th from the right = 22nd = V |
| kth to the right of the nth from the left | Position n + k | 8th to the right of the 14th from the left = 22nd = V |
| kth to the left of the nth from the left | Position n − k | 7th to the left of the 20th from the left = 13th = M |
Letter Pairs in a Word
"How many pairs of letters in the word have as many letters between them in the word as in the English alphabet?"
Compare every two letters: the number of letters between them in the word must equal the number between them in the alphabet. Unless the question says "forward direction only", count pairs in both directions.
Worked example — DEPOSIT (D E P O S I T):
| Pair | Letters between in the word | Letters between in the alphabet | Counts? |
|---|---|---|---|
| D–E | 0 (adjacent) | 0 (D = 4, E = 5) | Yes |
| P–O | 0 (adjacent) | 0 (P = 16, O = 15) | Yes |
| E–I | 3 (P, O, S) | 3 (F, G, H) | Yes |
| D–I | 4 (E, P, O, S) | 4 (E, F, G, H) | Yes |
| P–T | 3 (O, S, I) | 3 (Q, R, S) | Yes |
No other pair matches, so the answer is 5 pairs.
2. Meaningful Word Formation
"Using the 2nd, 3rd, 5th and 6th letters of MARKET, how many meaningful English words can be formed, using each letter only once?"
- Pick the letters by position from the original word: A (2nd), R (3rd), E (5th), T (6th).
- List arrangements that are common English words: RATE, TEAR, TARE.
- With options such as None / One / Two / More than two, the answer is More than two.
Tips:
- Take positions from the original word, never from a re-sorted version.
- Use each chosen letter exactly once.
- Write down every valid word before choosing; stopping at the first word leads to the "One" trap.
3. Digit and Symbol Condition Questions
Sequence: 5 2 7 3 8 1 9 4 6
"How many odd digits are immediately followed by an even digit?"
- 5 → 2 (yes), 7 → 3 (no), 3 → 8 (yes), 1 → 9 (no), 9 → 4 (yes).
- Answer: 3.
Scan for the rarer category first and check only its neighbours, exactly as in Section 2.1.
4. Calendar
Odd Days
An odd day is the remainder when a number of days is divided by 7.
| Period | Odd days |
|---|---|
| Ordinary year (365 days) | 1 |
| Leap year (366 days) | 2 |
| 100 years (76 ordinary + 24 leap) | 5 |
| 200 years | 3 |
| 300 years | 1 |
| 400 years | 0 |
| Month of 31 / 30 / 29 / 28 days | 3 / 2 / 1 / 0 |
Leap Year Rule
A year is a leap year if it is divisible by 4, except century years, which must be divisible by 400. So 2000 was a leap year, 1900 was not, and 2028 is.
Moving From a Known Date
- Year to year: 1 January 2026 was a Thursday. 2026 is an ordinary year (1 odd day), so 1 January 2027 is a Friday; 2027 is also ordinary, so 1 January 2028 is a Saturday.
- Month to month: 15 August 2026 is a Saturday. August has 31 days (3 odd days), so 15 September 2026 is a Tuesday.
- Known historical anchor: 15 August 1947 was a Friday.
When Does a Calendar Repeat?
A year's calendar repeats when the odd days in between add up to a multiple of 7 and both years are of the same type. From 2026 to 2036 the odd days total 14, so 2037 has the same calendar as 2026.
5. Clock
The Angle Formula
The minute hand moves 6° per minute and the hour hand 0.5° per minute, so at H hours and M minutes:
If θ is more than 180°, the smaller angle is 360° − θ.
- At 3:40: θ = |90 − 220| = 130°.
- At 7:20: θ = |210 − 110| = 100°.
When Do the Hands Coincide?
Between H and (H + 1) o'clock, the hands coincide at M = 60H ÷ 11 minutes past H.
- Between 4 and 5 o'clock: M = 240 ÷ 11 = 21 9/11, so the hands coincide at 4:21 9/11.
Frequency Facts
| Event | In 12 hours | In 24 hours |
|---|---|---|
| Hands coincide | 11 times | 22 times |
| Hands at right angles | 22 times | 44 times |
| Hands in a straight line, pointing opposite ways | 11 times | 22 times |
Mirror Images
For a time between 1:00 and 11:59, the mirror image = 11:60 − the given time. For example, 3:40 appears as 8:20 in a mirror.
Common Traps
| Trap | Correction |
|---|---|
| Counting letter pairs in one direction only | Count both directions unless told otherwise |
| Treating 1900 as a leap year | Century years need divisibility by 400 |
| Adding the full number of days instead of odd days | Reduce the day count modulo 7 first |
| Reporting the reflex clock angle | Give the smaller angle unless the question asks otherwise |
What is the smaller angle between the hour hand and the minute hand of a clock at 7:20?
If 1 January 2026 is a Thursday, what day of the week is 1 January 2028?
How many pairs of letters in the word BANKING have as many letters between them in the word as in the English alphabet, counting both directions?