Category Archives: Recreational mathematics

Playing card puzzles by Henry Dudeney


For the past couple of months, I have been publishing puzzles with playing cards. Henry Dudeney was British formost puzzle master of the late 19th / early 20th century. In this series his puzzles, as published in “Amusement in mathematics”, may not be omitted.

1) The card frame puzzle***/*****
In the illustration we have a frame constructed from the ten playing cards, ace to ten of diamonds. The children who made it wanted the pips on all four sides to add up alike, but they failed in their attempt and gave it up as impossible. It will be seen that the pips in the top row, the bottom row, and the left-hand side all add up 14, but the right-hand side sums to 23. Now, what they were trying to do is quite possible. Can you rearrange the ten cards in the same formation so that all four sides shall add up alike? Of course they need not add up 14, but any number you choose to select.


You can check your solution here

2) The cross of cards***/*****

In this case we use only nine cards—the ace to nine of diamonds. The puzzle is to arrange them in the form of a cross, exactly in the way shown in the illustration, so that the pips in the vertical bar and in the horizontal bar add up alike. In the example given it will be found that both directions add up 23. What I want to know is, how many different ways are there of rearranging the cards in order to bring about this result? It will be seen that, without affecting the solution, we may exchange the 5 with the 6, the 5 with the 7, the 8 with the 3, and so on. Also we may make the horizontal and the vertical bars change places. But such obvious manipulations as these are not to be regarded as different solutions. They are all mere variations of one fundamental solution. Now, how many of these fundamentally different solutions are there? The pips need not, of course, always add up 23.

You can check your solution here

3) The “T” card puzzle***/*****

An entertaining little puzzle with cards is to take the nine cards of a suit, from ace to nine inclusive, and arrange them in the form of the letter “T,” as shown in the illustration, so that the pips in the horizontal line shall count the same as those in the column. In the example given they add up twenty-three both ways. Now, it is quite easy to get a single correct arrangement. The puzzle is to discover in just how many different ways it may be done. Though the number is high, the solution is not really difficult if we attack the puzzle in the right manner. The reverse way obtained by reflecting the illustration in a mirror we will not count as different, but all other changes in the relative positions of the cards will here count. How many different ways are there?

You can check your solution here

4) Card triangles***/*****
Here you pick out the nine cards, ace to nine of diamonds, and arrange them in the form of a triangle, exactly as shown in the illustration, so that the pips add up the same on the three sides. In the example given it will be seen that they sum to 20 on each side, but the particular number is of no importance so long as it is the same on all three sides. The puzzle Pg 116is to find out in just how many different ways this can be done.

If you simply turn the cards round so that one of the other two sides is nearest to you this will not count as different, for the order will be the same. Also, if you make the 4, 9, 5 change places with the 7, 3, 8, and at the same time exchange the 1 and the 6, it will not be different. But if you only change the 1 and the 6 it will be different, because the order round the triangle is not the same. This explanation will prevent any doubt arising as to the conditions.

You can check your solution here

Playing card puzzles (4)


1) A 5×5 grid***/*****
Which of the 5 cards on the right should take the place of the card turned down?

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

Dice puzzles (3)


We had two previous posts on dice problems, which you can find here and here.

This is the third post in a small series on dice puzzles. The first one was about the polar bear puzzle and its variations, the second one posed some alternate patterns.
In this post I want to explore some dice puzzles by the British grand master of puzzles, Henry Dudeney.

1) The dice numbers.

I have a set of four dice, not marked with spots in the ordinary way, but with Arabic figures, as shown in the illustration. Each die, of course, bears the numbers 1 to 6. When put together they will form a good many, different numbers. As represented they make the number 1246. Now, if I make all the different four-figure numbers that are possible with these dice (never putting the same figure more than once in any number), what will they all add up to? You are allowed to turn the 6 upside down, so as to represent a 9. I do not ask, or expect, the reader to go to all the labour of writing out the full list of numbers and then adding them up. Life is not long enough for such wasted energy. Can you get at the answer in any other way?
This puzzles was published as puzzle 96 in “Amusement in Mathematics”

You can check your solutions here

2) A trick with dice

The first problem I want to have a look at was published by British puzzler Henry Dudeney in his book “Amusement in Mathematics” (and before that probably in one of the magazines in which he had a monthly column).
I ask you to throw three dice without me seeing them. Then I tell you to multiply the points of the first die by 2 and add 5. Multiply the result by 5 and add the point of the second die. Multiply the result by 10 and add the points on the third die. Mention me the result and I will immediately tell you the points on your three dice.
For example, if you throw 1, 3 and 6, the result will be 386, from which I could at once say what you had thrown. How do I do that?
This puzzles was published as puzzle 386 in “Amusement in Mathematics”

You can check your solutions here

3) The Montenegrin dice game
It is said that the inhabitants of Montenegro have a little dice game that is both ingenious and well worth investigation. The two players first select two different pairs of odd numbers (always higher than 3) and then alternately toss three dice. Whichever first throws the dice so that they add up to one of his selected numbers wins. If they are both successful in two successive throws it is a draw and they try again. For example, one player may select 7 and 15 and the other 5 and 13. Then if the first player throws so that the three dice add up 7 or 15 he wins, unless the second man gets either 5 or 13 on his throw.

The puzzle is to discover which two pairs of numbers should be selected in order to give both players an exactly even chance.

You can check your solutions here

Playing card puzzles (3)


This is the third post with puzzles about playing cards.

1) Which hands are valid?****/*****
In a game, the following six hands are valid plays:

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Which of the following three hands is / are valid?

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

Playing card puzzles (2)


Last month I published two puzzles with playing cards. Here are two more in the same line:

For those who missed the puzzles, look at the figure below. It shows 16 cards, one of which is hidden. At the bottom you find 4 cards. Which of those 4 cards should replace the hidden card?

3) Playing cards puzzle 3***/*****

You can check your solution here

4) Playing cards puzzle 4***/*****

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

Playing card puzzles (1)


While waiting for my appointment with the dentist I thumbed through the magazines, looking for good puzzles. I had little hope of finding any more than a crossword or sudoku, but to my surprise I encountered a new format in the magazine plusonline. The magazine does have a website, as the name suggests, but I couldn’t find the puzzles there.

In all puzzles the problem is: which of the four cards at the bottom should replace the blue card?

Here are two puzzles in the same vein. I did make a small change: the original puzzle had rows of three cards, which I changed into cards rows of four cards.

1) Playing cards square nr 1***/*****

You can check your solution here

2) Playing cards square nr 2***/*****

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

Be prepared for more puzzles of this type in a few weeks.

Dice puzzles old and new


Last month we had a look at the most famous of dice puzzles, the Polar bear puzzle. One of the beauties of dice puzzles is that, like playing cards, they are around in many bars.

I found an original dice puzzle in issue 70 of the long defunct British magazine Games & Puzzles, dated May/June 1978.

1) G&P issue 70***/*****



I’m not sure which one is older, the Polar Bears dice puzzle or the one in G&P.

You can check your solution here

2) 4 rolls of 4 dice***/*****



I’m not sure which one is older, the Polar Bears dice puzzle or the one in G&P.

You can check your solution here

Polar bears and other dice puzzles


Polar bears is no doubt the most famous dice puzzle around. I first heard it when I studied mathematics, and Douglas Hofstadters book “Godel, Escher, Bach” may have been the source.
If you want to puzzle your friends, roll 5 dice, and tell the how many polar bears can be spotted. Then roll 5 dice again, let them guess, and tell them the correct number if they guess wrong.

1) Polar bears***/*****
The polar bears puzzle is traditionally presented as a throw of 5 dice. If you are stumped, don’t despair, it is rumored that Bill Gates could only partially solve it.



Even though you may find it hard, I do encourage you to try to solve it before consulting the answer.

You can check your solution here

2) Seals***/*****
Polar bears hunt for seals. How many seals do you count?
This puzzle is inspired by the authors of https://www.pleacher.com/handley/puzzles/polrbear.html.


You can check your solution here

3) Fish***/*****
This puzzle too is inspired by the authors above, though in both instances I changed names to get a more logical picture.


You can check your solution here

Domino – lay out that set


Dutch puzzle designer Leon Balmaekers contacted me recently and told me he had written some booklets with puzzles for highly gifted children. The booklets are in Dutch, and contain a variety of puzzles. The highly gifted children in a classroom can make some of these puzzles when they have completed the normal exercises in a breeze.

One of the puzzle types uses a normal 0-6 domino set. Look at the figure in problem 1. In contrast to dominosa, the domino puzzle type most often used, the borders are clear, but the digits are missing.

Problem 1.**/*****
Domino_laydown_1_exercise
The numbers along the sides are the sum of the pips in the respective rows and columns. It is up to you to figure out which domino should go where. Normal domino rules are followed: whenever two bones lay end to end, the numbers are equal.

For your convenience, here is a complete double 6 set:
Domino_double_6_set

You can check your solution here

Problem 2**/*****
Domino_laydown_2_exercise

You can check your solution here

Problem 3***/*****
Domino_laydown_3_exercise

You can check your solution here

A new puzzle is published at least once a month on the first Friday of the month. Additional puzzles may be published on other Fridays.

Billy’s Big Christmas Party


Billy was delighted to have gained entry to the Big Christmas Party in the neighbouring village. There were stalls and booths littered throughout the hall.

1) The apple pie stall**/*****
The first stall he walked to showed delicious steaming hot pieces of apple pie on display. Several other kids had gathered in front of it.
A sign read:
4 and 7 give 33
3 and 2 give 5
8 and 5 give 39
A middle aged lady behind the stall held up two numbers: 6 and 3.
“This woman must be the wife of the math teacher,” he whispered to his neighbour.
The lady must have overheard him, because she laughed:
“Young man, I am the math teacher.”
But she was quickly satisfied when Billy quickly figured out the correct answer, collected his piece of apple pie and walked to the second stall.

You can check your solution here

2) The hot chestnuts stall****/*****
The second stall displayed dishes of chestnuts filled with had chestnuts. A man was roasting the chestnuts on a small coal fire and serving them with several sauces.
A sign displayed some calculations:
11 + 11 = 8
12 + 59 = 18
18 + 47 = 16
23 + 39 = 16
He held up two numbers for the children in front of his table: 22 and 45.
Slightly softer than the previous time, Billy whispered to the girl besides him:
“Do they have two math teachers here?”
The girl looked at him saying:
“Did you ask if we have two math teachers here?”
The man heard it and laughed: “No, I’m the Arts teacher.”
Billy quickly grasped the problem and found the sum of 22 and 45.

You can check your solution here

3) The mince pies****/*****
The third stall displayed a lovely looking plate with mince pies.
A piece of cardboard listed:
5 and 6 give 6
3 nd 7 give 7
7 and 8 give 8
She held up two cards showing 4 and 12. “What number do they give?” she asked. “I’ll tell you in advance the answer is not 12.”
To prevent him from asking, the girl besides him told him:
“No, she doesnt teach math. She teaches English.”

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.