# Five further sequence puzzles

This is my second batch of sequence puzzles that are nasty and hard to solve; yet, each of them has a clear and justifiable solution.

Sequence 1:
18, 17, 18, 22, 26, 35, 32, 32, ??, 38, ??, 52, ??, ??, ??, 64, ...

Sequence 2:
Du, Me, Ph, Me, Al, Mi, ??

Sequence 3:
..., 14, 13, 14, 6, 7, 12, 8, 16, 9, 13, 10, 15, 11, 12, ??, 6, ??, 2, ??, ??

Sequence 4:
1, 2, 3, 3, 2, 3, 4, 5, 3, 2, 3, 4, 5, 5, 4, ?, ?, ?, ...

Sequence 5:
?, ?, ?, ?, M, A, J, ?, H, T, P, T, F, P, B, L, J, G, H, G, A, C, H, C, ?, R, T, W, H, C, ...

• You guys are really fast! I thought these would take much longer to solve... Feb 4, 2015 at 20:14
• Since sequence 2 has been solved last, I count it as the most difficult one of the five sequences and I assign the "correct answer" sign to Julian Rosen's answer. Feb 5, 2015 at 19:19
• I made a CW answer combining all of the solutions. It might be more useful to have that as the accepted answer instead. Feb 5, 2015 at 22:57

Sequence 1

Number of matches played in the FIFA World Cups:
18, 17, 18, 22, 26, 35, 32, 32, 32, 38, 38, 52, 52, 52, 52, 64,...
Source

Sequence 2

Stars in the Big Dipper (the seven brightest stars in the constellation Ursa Major):
Dubhe, Merak, Phecda, Megrez, Alioth, Mizar, Alkaid
Source

Sequence 3

Popes:
..., Benedict XIV (14), Clement XIII (13), Clement XIV (14), Pius VI (6), Pius VII (7), Leo XII (12), Pius VIII (8), Gregory XVI (16), Pius IX (9), Leo XIII (13), Pius X (10), Benedict XV (15), Pius XI (11), Pius XII (12), John XXIII (23), Paul VI (6), John Paul I (1), John Paul II (2), Benedict XVI (16), Francis I (1)
Source

Sequence 4

Number of strokes needed to write Roman numerals:
I (1), II (2), III (3), IV (3), V (2), VI (3), VII (4), VIII (5), IX (3), X (2), XI (3), XII (4), XIII (5), XIV (5), XV (4), XVI (5), XVII (6), XVIII (7),...

Sequence 5

Presidents of the United States:
Washington, Adams, Jefferson, Madison, Monroe, Adams, Jackson, Van Buren, Harrison, Tyler, Polk, Taylor, Fillmore, Pierce, Buchanan, Lincoln, Johnson, Grant, Hayes, Garfield, Arthur, Cleveland, Harrison, Cleveland, McKinley, Roosevelt, Taft, Wilson, Harding, Coolidge,...
Source

Sequence 2 is

the initial two letters of the names of the stars in the Big Dipper (the seven brightest stars in the constellation Ursa Major). The stars are:
Dubhe
Merak
Phecda
Megrez
Alioth
Mizar
Alkaid
The missing letters are 'Al'.

I first noticed that sequence 2 was much shorter than the others, having only seven items. I did a Google search for 'things that come in sevens', and one of the lists mentioned there were seven stars in the Big Dipper. I looked up the names of the stars, and found they were a match.

Sequence 1:

Number of matches played in the FIFA World Cups since 1930. Missing entries are 32, 38, 52, 52, 52. Source

Sequence 5 is

Presidents of the United States.
W(ashington), A(dams), J(efferson), M(adison), M(onroe), A(dams), J(ackson), V(an Buren), H(arrison), T(yler), P(olk), T(aylor), F(illmore), P(ierce), B(uchanan), L(incoln), J(ohnson), G(rant), H(ayes), G(arfield), A(rthur), C(leveland), H(arrison), C(leveland), M(cKinley), R(oosevelt), T(aft), W(ilson), H(arding), C(oolidge).
The sequence ends here, but the presidents continue: H(oover), R(oosevelt), T(ruman), E(isenhower), K(ennedy), J(ohnson), N(ixon), F(ord), C(arter), R(eagan), B(ush), C(linton), B(ush), O(bama).
The missing letters are W, A, J, M, V, M.

• How did you figure this out? Feb 4, 2015 at 17:55
• Pretty much a lucky guess. Feb 4, 2015 at 17:56

Sequences 1, 3, and 4 again can be googled:

Sequence 1:

Goles totales in mismunidales = Number of goals scored at the Soccer World Championships
18 17 18 22 26 35 32 32 32 38 38 52 52 64 64 64

Sequence 3:

The numbering of the popes,
ending with John XXIII, Paul VI, John Paul I, John Paul 2, Benedict XVI, Francis I:
..., 14, 13, 14, 6, 7, 12, 8, 16, 9, 13, 10, 15, 11, 12, 23, 6, 1, 2, 16, 1

Sequence 4:

number of strokes in the Roman numerals: I, II, III, IV, ...
1, 2, 3, 3, 2, 3, 4, 5, 3, 2, 3, 4, 5, 5, 4, 5, 6, 7

• BTW, your interpretation of sequence 1 is wrong.
– dmg
Feb 4, 2015 at 20:18
• @dmg: I see. Google showed me the number sequence on some page in Spanish, and I only understood that the numbers are somehow from the soccer world cup. Feb 4, 2015 at 20:23
• Same here. Just crosschecked in Wikipedia what exactly about the world cup did those number say :D
– dmg
Feb 4, 2015 at 20:27