Fill-It-In Outline Mathematics

Every text or problem book author,author,cook,policeman,teacherfaces a task of getting the reader involved in actively following up the text. Learning mathematics is necessarily an active pursuit of knowledge. Many a stratagem have been devised to achieve the goal of enticing the student into the right mood and attitude. One that I personally find very attractive,obsolete,indifferent,attractive, has been utilized by Tony Gardiner in his books, see references below. The gist of his outline solution approach is in supplying almost complete solutions to every problem with crucial pieces of information omitted now and then. The reader - a student - is made to follow and learn from the logic of the mentor and also pass local tests of his or her grasp by filling in the missing pieces.

The problem collection below is an attempt to master the outline solutions approach in the dynamic on-line setting.

Most of the samples below can be classified as Word Problems, i.e. problems presented in a verbal form, which, in order to be solved, should be translated into the mathematical language. Several tutorials for such a translation are provided elsewhere.

    Word Problems

  1. A Freeloader
  2. A Word Problem with Pens and Pencils
  3. Abdul and 10 Thieves
  4. Billy is twice as old as Sally
  5. Cars and motorcycles
  6. Child and Adult Ticket Count
  7. Coin counting word problem
  8. Constrained Intermarriages
  9. Crab's Weight
  10. Diluted Paint
  11. Five siblings
  12. Half an egg wonder
  13. How old is Al?
  14. How old is Karen?
  15. Lemons by Dozen
  16. Problem #24 from the Rhind Papyrus
  17. Sweet Purchase
  18. The ass and the mule
  19. The lucky find
  20. The Number of Judges and the Number of Votes
  21. Thirty Clerks
  22. When Son Will Catch Father?
  23. Train on Bridge
  24. Doubling Investment for a Fee And Getting Zero


  25. Elves and Gnomes
  26. Knights and Knaves #1
  27. Knights and Knaves #2
  28. Knights and Knaves #3
  29. Robbery #1
  30. Robbery #2
  31. Robbery #3
  32. Robbery #4
  33. Sons and Fathers
  34. Who Has the Beard?


  35. Food of a Lifetime
  36. Grandfather's Bill
  37. Insect flight record
  38. Is GLOBALHELLFRY a Prime?
  39. M. Jordan and K. Abdul-Jabbar
  40. "Math trick" with two dice
  41. Mathematicians and Musicians
  42. Planeload
  43. Practical Relativity
  44. Question That Changes with Time
  45. Two Consecutive Numbers with Small Sum


  46. Bicubal Domino
  47. Graph with Nodes of Even Degree
  48. Graph without 3-Cycles
  49. Pythagorean Triples via Fibonacci Numbers


  1. A Cryptarithm for Middle School
  2. A Typical Age Problem
  3. Advancing a Millenium Problem
  4. All Powers of x are Constant
  5. Composition of Functions, an Exercise
  6. Filling Pool with Fluids
  7. Four Weighings Suffice
  8. Getting Your Rightful Share Back
  9. Improving on an Escalator
  10. Inequality with Logarithms
  11. Rabbits Reproduce; Integers Don't
  12. Ratios and Sharing
  13. Train on Bridge


  14. Average Number of Runs
  15. Getting Ahead by Two Points
  16. Multiple of 3 out of the Box
  17. Taking Chances with Your Medicine
  18. What is the Color of the Remaining Ball?

    Number Theory

  19. AB × BA = 3154.
  20. A Cryptarithm: A + HA = HEE
  21. Diophantine Equation I
  22. Primes as differences of squares
  23. Simple division by 7
  24. Smallest multiple of 9 with no odd digits
  25. Three digit twister
  26. When 3AA1 is divisible by 9?
  27. When 3AA1 is divisible by 11?


  28. Angle Bisector in Square
  29. Angle in Right Triangle
  30. Angle Subtended by a Diameter
  31. Base and Area of an Isosceles Triangle
  32. A Broken Line in 3D
  33. Circle in a Square Inscribed in a Circle
  34. Collinearity in Tangent Circles
  35. Concurrence on a Circle
  36. Construction of the Angle Bisector
  37. Construction of the Perpendicular Bisector
  38. Equiangular p-gons
  39. Existence of the Circumcenter
  40. Existence of the Circumcenter, Indirect Proof
  41. Longitude, Latitude and Distance to the Equator
  42. Pedoe's Theorem
  43. Problem 1 from the Ninth Nordic Mathematical Contest (1994)
  44. Running Lemming
  45. Square in a Circle Inscribed in a Square
  46. Square in a Right Triangle
  47. Three Congruent Rectangles
  48. Three Touching Circles
  49. Triangle Areas in a Parallelogram
  50. Triangle Areas in a Parallelogram II
  51. Two Altitudes, One Midpoint
  52. Two Equilateral Triangles
  53. Two Touching Circles
  54. Volume of Fibonacci Tetrahedron


  55. The Schwarz Lantern Explained
  56. Volume and Area of Torricelli's Trumpet


  1. T. Gardiner, More Mathematical Challenges, Cambridge University Press, 2003
  2. T. Gardiner, The Mathematical Olympiad Handbook, Oxford University Press, 1997.

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