mathschallenge.net

3x3 Magic Square      Problem ID: 366 (15 Nov 2009)

By considering rotations and reflections to be equivalent, prove that there exists only one 3x3 magic square.

Adding Digits      Problem ID: 109 (Mar 2003)

If every combination of the digits 1,2,3,4 was written down, what would be the sum of the numbers formed?

Alternate Fibonacci Ratio      Problem ID: 310 (15 Feb 2007)

What does F_$n$+2/F_$n$ tend towards as $n$ increases?

Alternating Squares      Problem ID: 119 (May 2003)

Can you work out what fraction of the diagram is shaded?

Annulus      Problem ID: 159 (Mar 2004)

Find the width of the annulus

Area Of Annulus      Problem ID: 341 (26 Jun 2008)

Find the area of the annulus.

Average Matches      Problem ID: 303 (12 Jan 2007)

Is the claim about the average contents of matches in a matchbox correct?

A Number And Its Reciprocal      Problem ID: 297 (17 Dec 2006)

Prove that $x$ + 1/$x$ ≥ 2 for non-negative values of $x$.

Blonde Hair Brown Eyes      Problem ID: 145 (Jan 2004)

What fraction of the class in total have brown eyes?

Box World      Problem ID: 55 (Nov 2001)

How many unique four unit cube arrangements are there?

Calendar Cubes      Problem ID: 103 (Feb 2003)

Show how you would label two wooden cubes to display any date of the month.

Candelabra      Problem ID: 183 (01 Nov 2004)

Find the probability that the three white candles will be adjacent

Chequered Floor      Problem ID: 87 (Nov 2002)

Can you discover the dimensions of the room?

Christmas Trees      Problem ID: 52 (Oct 2001)

Can you work out the most efficient way to plant trees?

Circular Pipes      Problem ID: 36 (Feb 2001)

Using a metre stick how would you find the internal diameter of a large circular pipe?

Circumscribed Triangle      Problem ID: 108 (Mar 2003)

Can you find the area of the circumscribed triangle?

Climbing Stairs      Problem ID: 74 (Apr 2002)

How many ways can you climb ten steps?

Coloured Dice      Problem ID: 124 (Oct 2003)

How should the 2nd die be coloured so that there is an equal chance of getting two faces of the same colour?

Consecutive Cube      Problem ID: 83 (Oct 2002)

Can you prove that the product of three consecutive integers, plus their mean, is always cube?

Consecutive Prime Sum      Problem ID: 332 (19 Nov 2007)

Show that two consecutive primes cannot have a sum that is double a prime.

Counting Sequence      Problem ID: 207 (17 Feb 2005)

Can you determine the 1000^th term of the sequence?

Curvature Of The Earth      Problem ID: 44 (Apr 2001)

How far can you see from the top of the Eiffel tower?

Disco Ratios      Problem ID: 131 (Nov 2003)

Can you work out how many girls are at the disco?

Divided Square      Problem ID: 339 (18 Jun 2008)

Prove that the only number of non-overlapping squares you cannot split a unit square into are 2, 3, or 5 smaller squares.

Dodecagon Edges      Problem ID: 73 (Apr 2002)

How many edges does a dodecagon have?

Dotty Squares      Problem ID: 35 (Feb 2001)

How many squares can you draw on a grid measuring 4 dots by 4 dots?

Double An Odd Sum      Problem ID: 283 (23 Jul 2006)

If $S_n$ represents the sum of the first $n$ odd numbers, prove that $4S_n = S_{2n}$.

Egyptian Divisibility      Problem ID: 112 (Apr 2003)

What is the least number which has no remainder when divided by any number from 1 to 10?

Equal Angles      Problem ID: 196 (21 Dec 2004)

Prove that the two angles in the square are congruent.

Expressing Divisibility      Problem ID: 356 (26 Aug 2009)

Prove that $n$($n$ + 1)(2$n$ + 1) is divisible by six for all integer values, $n$.

Factorial Symmetry      Problem ID: 214 (09 Mar 2005)

Solve the equation a!b! = a! + b!

Favourite Mathematician      Problem ID: 4 (Aug 2000)

Can you work out the teacher's favourite mathematician?

Finishing On 150      Problem ID: 39 (Mar 2001)

How many ways can a darts player finish from 150 points?

Five-digit Divisibility      Problem ID: 199 (10 Jan 2005)

Using the digits 1, 2, 3, 4, and 5 to form 5-digit numbers, how many are divisible by 12?

Four Hats      Problem ID: 337 (16 May 2008)

Which of the boys, 1 to 3, is most likely to guess the colour of his hat?

Fractional Steps      Problem ID: 349 (30 Nov 2008)

Can you determine the best way to construct a set of steps leading up to a platform?

Glass In The Door      Problem ID: 165 (Apr 2004)

Can you determine the weight of the glass in the door?

Half Fractions      Problem ID: 151 (Feb 2004)

By concatenating all of the digits 1, 2, 3, 4, and 5 to form the ratio of two numbers, how many ways can you make one-half?

Hexagon Perimeter      Problem ID: 278 (13 May 2006)

Find the perimeter of the hexagon.

Hockey Lockers      Problem ID: 43 (Apr 2001)

From the clues can you work out how many lockers there are?

Ice Cream Cone      Problem ID: 71 (Mar 2002)

Can you find the height of the cone?

Inscribed Circle In Isosceles Triangle      Problem ID: 375 (16 Aug 2010)

Find the radius of the circle inscribed inside the isosceles triangle.

Inscribed Circle In Right Angled Triangle      Problem ID: 364 (03 Nov 2009)

Find the radius of the circle inscribed inside the right angled triangle.

Inscribed Square      Problem ID: 368 (30 Nov 2009)

Find the side length of the square inscribed inside the right angled triangle.

Integer Fraction Product      Problem ID: 114 (Apr 2003)

Can you show when the product of fractions is an integer?

Largest Root      Problem ID: 345 (21 Sep 2008)

Which is greater in value, the square root of two or the cube root of three?

Leaning Ladders      Problem ID: 113 (Apr 2003)

How far above the ground do the two ladders cross?

Letter Product      Problem ID: 88 (Nov 2002)

Can you find the exact value of the letter product?

Lucky Guess      Problem ID: 138 (Dec 2003)

Find the probability of winning the card game

Maximum Square      Problem ID: 152 (Feb 2004)

What are the dimensions of the shaded square inside the 3-4-5 right angle triangle?

Meaning Of Life      Problem ID: 6 (Aug 2000)

What is the password to unlock all secrets?

Measuring Mountains      Problem ID: 132 (Nov 2003)

Use the given information to find the height of the mountain.

Missing Weight      Problem ID: 66 (Feb 2002)

Can you work out the missing weight?

Modest Age      Problem ID: 130 (Nov 2003)

Can you work out how old the teacher really is?

Mystic Rose      Problem ID: 93 (Dec 2002)

How many lines are required to construct the Mystic Rose?

Net Perimeter      Problem ID: 360 (11 Oct 2009)

By considering all the nets of a unit cube, which net has the greatest perimeter?

Never Divides By 5      Problem ID: 318 (07 Apr 2007)

Prove that $x$² + $x$ + 1 will never divide by 5.

Numbered Discs      Problem ID: 157 (Mar 2004)

Can you work out which numbers are on the two discs?

Numbered Discs 2      Problem ID: 164 (Apr 2004)

If you had discs numbered 1 to 10, how would you separate the discs into the two bags such that no bag contains its double?

Painted Cubes      Problem ID: 51 (Oct 2001)

How many ways can paint a cube with two colours?

Password Cracker      Problem ID: 7 (Aug 2000)

Can you crack the hacker website password?

Pen Problem      Problem ID: 99 (Jan 2003)

Can you work out how many pens the girl bought and how much she paid for each one?

Perfect Cone      Problem ID: 203 (24 Jan 2005)

Can you find the height of the cone?

Perfect Ruler      Problem ID: 70 (Mar 2002)

What is the minimum number of marks required to measure the lengths 1 to 12?

Permuted Sums      Problem ID: 211 (06 Mar 2005)

Find the sum of all 4-digit combinations taken from {1,2,3,4,5}

Prime One Less Than Square      Problem ID: 281 (15 Jul 2006)

Finding primes that are one less than a square.

Prime Square Differences      Problem ID: 94 (Dec 2002)

Can you prove that all primes greater than 2 can be written as the difference of two squares?

Prime Square Divisibility      Problem ID: 67 (Feb 2002)

Can you prove that the square of all primes minus 1 are divisible by 24?

Prime Square Sums      Problem ID: 123 (Oct 2003)

How many primes less than 100 can be written as the sum of two square numbers?

Prime Uniqueness      Problem ID: 48 (May 2001)

Prove that seven is the only prime number that is one less than a perfect cube.

Proportional Difference      Problem ID: 314 (18 Mar 2007)

Show that $a/b = c/d = (a - c)/(b - d)$.

Proportion Of Ones      Problem ID: 189 (28 Nov 2004)

What proportion of 3-digit numbers contain the digit one?

Quadrant Product Divisibility      Problem ID: 222 (24 May 2005)

How would you arrange the numbers 1 to 16 in the grid, such that the product of the numbers in each quadrant is divisible by 16?

Random Routes      Problem ID: 172 (May 2004)

How many different routes can you find through a 4 × 4 grid?

Rectangle Construction      Problem ID: 376 (17 Oct 2010)

Find the connection between the constructed length and the original rectangle.

Rectangular Circles      Problem ID: 253 (12 Dec 2005)

Given the three concentric circles generated by the rectangle show that the area of the inner circle equals the ring generated by the outer circles

Remainder Of One      Problem ID: 233 (31 Jul 2005)

Find the smallest number, greater than 1, which has a remainder of 1 when divided by any of 2, 3, 4, 5, 6, or 7

Reverse Digits      Problem ID: 82 (Oct 2002)

Investigating the divisibility of adding a 2-digit number to its reverse.

Right Time      Problem ID: 137 (Dec 2003)

How many times between 9 a.m. and 3 p.m. is the angle between the hour and minute hand 90^o?

Right Triangle Equal Angles      Problem ID: 248 (27 Nov 2005)

In the given right angle triangle prove that the two marked angles are the same size

Rounded Roots      Problem ID: 47 (May 2001)

Investigate the special number machine that square roots and rounds off answers.

Rounding Error      Problem ID: 343 (09 Jul 2008)

Finding the chance of making a rounding error.

Rounding Machine      Problem ID: 97 (Jan 2003)

Can you discover the connection between the input values produce the same output from two different machines?

Running Requirements      Problem ID: 268 (11 Feb 2006)

What should the length of the straight section be on a running track be to meet IAAF requirements?

Same Digits      Problem ID: 229 (10 Jul 2005)

How many 3-digit numbers have two digits the same?

Same Digit Prime      Problem ID: 256 (01 Jan 2006)

Explain why a number made up of the same digit can only be prime if the digit is one AND the number of digits is itself prime

Sand Glass      Problem ID: 226 (04 Jun 2005)

Using one sand glass that measures 9 minutes and another that measures 13 minutes, how would you measure 30 minutes?

Save Rate      Problem ID: 102 (Feb 2003)

From the information given, how can the goal keeper achieve a 50% save rate?

Semi-circle Lunes      Problem ID: 173 (May 2004)

Find the area of the lunes on the semi-circle

Shaded Cross      Problem ID: 125 (Oct 2003)

Find the area of the shaded cross.

Shaded Triangle      Problem ID: 98 (Jan 2003)

Can you find the area of the shaded square inside the triangle?

Simple Fractions Symmetry      Problem ID: 166 (Apr 2004)

For a given denominator, prove that there are always an even number of simple fractions

Slide Height      Problem ID: 294 (26 Nov 2006)

Find the vertical height of the slide

Sloping Square      Problem ID: 179 (Oct 2004)

Find the area of the sloping square.

Snapped Pole      Problem ID: 362 (28 Oct 2009)

Can you determine the point where the pole snapped?

Solid Encryption      Problem ID: 9 (Aug 2000)

Can you decrypt the punchline?

Square And Round Plugs      Problem ID: 370 (24 Dec 2009)

Which fits better... a round plug in a square hole or a square plug in a round hole?

Square Laminas      Problem ID: 56 (Nov 2001)

How many squares can you make from 240 unit tiles?

Square Product      Problem ID: 178 (Oct 2004)

Given that [n(n+1)(n+2)]² = 3039162537*6, find the value of *

Square Rods      Problem ID: 188 (28 Nov 2004)

Using rods of length 1, 2, 3, ... , n, can you construct a square?

String Of Ones      Problem ID: 287 (09 Sep 2006)

Find the smallest repunit that is divisible by 63.

Taming The Sum      Problem ID: 140 (Dec 2003)

Can you make the sum equal to 100?

Tangential Distances      Problem ID: 351 (17 Apr 2009)

Prove that the tangential distances PS and PT are always equal.

Three Squares      Problem ID: 317 (07 Apr 2007)

Prove that the sum of the given angles are 360 degrees.

Tiled Floor      Problem ID: 54 (Nov 2001)

Can you work out how many tiles fill the room?

Triangle Area      Problem ID: 239 (10 Aug 2005)

Given two points, X and Y, find the area of triangle OXY

Tri Angles      Problem ID: 243 (19 Oct 2005)

Prove that exterior angle of a triangle is equal to the sum of two opposite interior angles

Tunnel Train      Problem ID: 104 (Feb 2003)

How long will it take for the train to completely pass through the tunnel?

Unsorted Socks      Problem ID: 358 (27 Sep 2009)

Find the minimum number of socks which must be taken from the drawers to be certain of finding five matching pairs.

Up Down Left Right      Problem ID: 11 (Aug 2000)

Can you discover the secret of this misleading system?

Weighty Logic      Problem ID: 218 (30 Mar 2005)

Can you use a combination of logic and algebra to determine Goldilock's weight?

X Hits The Spot      Problem ID: 12 (Aug 2000)

Can you discover what X represents?